[Update] Techniques for Adding the Numbers 1 to 100 – BetterExplained | 1+2+3+4+5+6 – NATAVIGUIDES

There’s a popular story that Gauss, mathematician extraordinaire, had a lazy teacher. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100.

Gauss approached with his answer: 5050. So soon? The teacher suspected a cheat, but no. Manual addition was for suckers, and Gauss found a formula to sidestep the problem:

Let’s share a few explanations of this result and really understand it intuitively. For these examples we’ll add 1 to 10, and then see how it applies for 1 to 100 (or 1 to any number).

Technique 1: Pair Numbers

Pairing numbers is a common approach to this problem. Instead of writing all the numbers in a single column, let’s wrap the numbers around, like this:

1  2  3  4  5
10 9  8  7  6

An interesting pattern emerges: the sum of each column is 11. As the top row increases, the bottom row decreases, so the sum stays the same.

Because 1 is paired with 10 (our n), we can say that each column has (n+1). And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs.

which is the formula above.

Wait — what about an odd number of items?

Ah, I’m glad you brought it up. What if we are adding up the numbers 1 to 9? We don’t have an even number of items to pair up. Many explanations will just give the explanation above and leave it at that. I won’t.

Let’s add the numbers 1 to 9, but instead of starting from 1, let’s count from 0 instead:

0  1  2  3  4
9  8  7  6  5

By counting from 0, we get an “extra item” (10 in total) so we can have an even number of rows. However, our formula will look a bit different.

Notice that each column has a sum of n (not n+1, like before), since 0 and 9 are grouped. And instead of having exactly n items in 2 rows (for n/2 pairs total), we have n + 1 items in 2 rows (for (n + 1)/2 pairs total). If you plug these numbers in you get:

which is the same formula as before. It always bugged me that the same formula worked for both odd and even numbers – won’t you get a fraction? Yep, you get the same formula, but for different reasons.

Technique 2: Use Two Rows

The above method works, but you handle odd and even numbers differently. Isn’t there a better way? Yes.

Instead of looping the numbers around, let’s write them in two rows:

1  2  3  4  5  6  7  8  9  10
10 9  8  7  6  5  4  3  2  1

Notice that we have 10 pairs, and each pair adds up to 10+1.

The total of all the numbers above is

But we only want the sum of one row, not both. So we divide the formula above by 2 and get:

Now this is cool (as cool as rows of numbers can be). It works for an odd or even number of items the same!

Technique 3: Make a Rectangle

I recently stumbled upon another explanation, a fresh approach to the old pairing explanation. Different explanations work better for different people, and I tend to like this one better.

Instead of writing out numbers, pretend we have beans. We want to add 1 bean to 2 beans to 3 beans… all the way up to 5 beans.

x
x x
x x x
x x x x
x x x x x

Sure, we could go to 10 or 100 beans, but with 5 you get the idea. How do we count the number of beans in our pyramid?

Well, the sum is clearly 1 + 2 + 3 + 4 + 5. But let’s look at it a different way. Let’s say we mirror our pyramid (I’ll use “o” for the mirrored beans), and then topple it over:

x                 o      x o o o o o
x x             o o      x x o o o o
x x x         o o o  =>  x x x o o o
x x x x     o o o o      x x x x o o
x x x x x o o o o o      x x x x x o

Cool, huh? In case you’re wondering whether it “really” lines up, it does. Take a look at the bottom row of the regular pyramid, with 5′x (and 1 o). The next row of the pyramid has 1 less x (4 total) and 1 more o (2 total) to fill the gap. Just like the pairing, one side is increasing, and the other is decreasing.

Now for the explanation: How many beans do we have total? Well, that’s just the area of the rectangle.

We have n rows (we didn’t change the number of rows in the pyramid), and our collection is (n + 1) units wide, since 1 “o” is paired up with all the “x”s.

Notice that this time, we don’t care about n being odd or even – the total area formula works out just fine. If n is odd, we’ll have an even number of items (n+1) in each row.

But of course, we don’t want the total area (the number of x’s and o’s), we just want the number of x’s. Since we doubled the x’s to get the o’s, the x’s by themselves are just half of the total area:

And we’re back to our original formula. Again, the number of x’s in the pyramid = 1 + 2 + 3 + 4 + 5, or the sum from 1 to n.

Technique 4: Average it out

We all know that

average = sum / number of items

which we can rewrite to

sum = average * number of items

So let’s figure out the sum. If we have 100 numbers (1…100), then we clearly have 100 items. That was easy.

To get the average, notice that the numbers are all equally distributed. For every big number, there’s a small number on the other end. Let’s look at a small set:

1 2 3

The average is 2. 2 is already in the middle, and 1 and 3 “cancel out” so their average is 2.

For an even number of items

1 2 3 4

the average is between 2 and 3 – it’s 2.5. Even though we have a fractional average, this is ok — since we have an even number of items, when we multiply the average by the count that ugly fraction will disappear.

Notice in both cases, 1 is on one side of the average and N is equally far away on the other. So, we can say the average of the entire set is actually just the average of 1 and n: (1 + n)/2.

Putting this into our formula

And voila! We have a fourth way of thinking about our formula.

So why is this useful?

Three reasons:

1) Adding up numbers quickly can be useful for estimation. Notice that the formula expands to this:

Let’s say you want to add the numbers from 1 to 1000: suppose you get 1 additional visitor to your site each day – how many total visitors will you have after 1000 days? Since thousand squared = 1 million, we get million / 2 + 1000/2 = 500,500.

2) This concept of adding numbers 1 to N shows up in other places, like figuring out the probability for the birthday paradox. Having a firm grasp of this formula will help your understanding in many areas.

3) Most importantly, this example shows there are many ways to understand a formula. Maybe you like the pairing method, maybe you prefer the rectangle technique, or maybe there’s another explanation that works for you. Don’t give up when you don’t understand — try to find another explanation that works. Happy math.

By the way, there are more details about the history of this story and the technique Gauss may have used.

Variations

Instead of 1 to n, how about 5 to n?

Start with the regular formula (1 + 2 + 3 + … + n = n * (n + 1) / 2) and subtract off the part you don’t want (1 + 2 + 3 + 4 = 4 * (4 + 1) / 2 = 10).

Sum for 5 + 6 + 7 + 8 + … n = [n * (n + 1) / 2] – 10

And for any starting number a:

Sum from a to n = [n * (n + 1) / 2] – [(a - 1) * a / 2]

We want to get rid of every number from 1 up to a – 1.

How about even numbers, like 2 + 4 + 6 + 8 + … + n?

Just double the regular formula. To add evens from 2 to 50, find 1 + 2 + 3 + 4 … + 25 and double it:

Sum of 2 + 4 + 6 + … + n = 2 * (1 + 2 + 3 + … + n/2) = 2 * n/2 * (n/2 + 1) / 2 = n/2 * (n/2 + 1)

So, to get the evens from 2 to 50 you’d do 25 * (25 + 1) = 650

How about odd numbers, like 1 + 3 + 5 + 7 + … + n?

That’s the same as the even formula, except each number is 1 less than its counterpart (we have 1 instead of 2, 3 instead of 4, and so on). We get the next biggest even number (n + 1) and take off the extra (n + 1)/2 “-1″ items:

Sum of 1 + 3 + 5 + 7 +  … + n = [(n + 1)/2 * ((n + 1)/2 + 1)] – [(n + 1) / 2]

To add 1 + 3 + 5 + … 13, get the next biggest even (n + 1 = 14) and do

[14/2 * (14/2 + 1)] – 7 = 7 * 8 – 7 = 56 – 7 = 49

Combinations: evens and offset

Let’s say you want the evens from 50 + 52 + 54 + 56 + … 100. Find all the evens

2 + 4 + 6 + … + 100 = 50 * 51

and subtract off the ones you don’t want

2 + 4 + 6 + … 48 = 24 * 25

So, the sum from 50 + 52 + … 100 = (50 * 51) – (24 * 25) = 1950

Phew! Hope this helps.

Ruby nerds: you can check this using

(50..100).select {|x| x % 2 == 0 }.inject(:+)
1950

Javascript geeks, do this:

[...Array(51).keys()].map(x => x + 50).filter(x => x % 2 == 0).reduce((x, y) => x + y)
1950

// Note: There are 51 numbers from 50-100, inclusive. Fencepost!

Other Posts In This Series

Difference between wifi 6 and wifi 5,wifi 4,wifi 3,wifi 2,wifi 1

This page compares wifi 6 vs wifi 5 vs wifi 4 vs wifi 3 vs wifi 2 vs wifi 1 and mentions
difference between wifi 6, wifi 5, wifi 4, wifi 3, wifi 2 and wifi 1 with respect to
speed, range and other parameters.

Introduction:
LAN is used to provide wired connectivity to computing devices with the help of
ethernet cable. As LAN does not support wireless connectivity, WLAN has been developed.
WLAN is also known as Wireless Local Area Network or WiFi (Wireless Fidelity).
In order to establish WiFi network, WiFi router is essential.
WiFi router connects one side with wired broadband connection and provide wireless signals to the
other side for wireless devices such as laptops, smartphones and tablets.

IEEE WLAN group has developed series of WiFi standards to support various speeds, ranges, frequencies.
The series include 802.11b, 802.11a, 802.11g, 802.11n, 802.11ac and 802.11ax.
The same series of WLAN standards can be designated as WiFi-1 to WiFi-6 respectively.

WiFi 1

The Standard IEEE 802.11b is referred as WiFi 1.
This is the first WiFi standard developed by IEEE for Wireless LAN.
This WiFi 1 devices operate at 2.4 GHz frequency band and uses DSSS/CCK modulation schemes for data.
It provides support for different speed as per mod-code rate viz. 1, 2, 5.5 and 11 Mbps.
WiFi-1 supports coverage distances of about 38 meters in indoor and about 140 meters in outdoor environments.
Refer CCK vs DSSS vs OFDM>>
for more information.

WiFi 2

The Standard IEEE 802.11a is referred as WiFi 2.
This WiFi Standard is successor to IEEE 802.11b (i.e. WiFi 1).
This is the first wifi standard in which multi carrier modulation scheme i.e. OFDM has been
introduced to support high data rates unlike single carrier used in wifi-1.
This WiFi-2 version based routers (i.e. APs) and Stations (i.e. clients) operate at 5 GHz RF carrier frequency.
This standard based devices support various speed or data rates such as 6, 9, 12, 18, 24, 36, 48 and 54 Mbps due to
use of 20 MHz bandwidth. WiFi-2 compliant routers support distance coverage or range of
about 35 meters (in indoors) and 120 meters (in outdoors).

WiFi 3

The Standard IEEE 802.11g is referred as WiFi 3.
This WiFi Standard is successor to IEEE 802.11a (i.e. WiFi 2).
The WiFi-3 or 11g standard has been developed to support frequency bands of both 11b and 11a standard
based devices. It supports 2.4 GHz and 5 GHz frequency bands.
It supports all the features as supported by 11a and 11b standards.

WiFi 4

The Standard IEEE 802.11n is referred as WiFi 4.
This WiFi Standard is successor to IEEE 802.11g (i.e. WiFi 3).
This is the wifi standard in which MIMO has been introduced.
Beamforming has been introduced but interoperabilities have not been tested.
It supports legacy fallbacks to previous wifi versions viz. wifi-1, wifi-2 and wifi-3.
It supports 20 MHz and 40 MHz bandwidths. Due to use of MIMO and higher BW (i.e. 40 MHz)
data rates up to 150Mbps can be achieved. WiFi-4 devices can support range of about 70 meters in indoor and
about 250 meters in outdoor environments. MIMO configurations supported by WiFi-4 devices include 2T3R and 4T4R.
Modulation schemes such as BPSK, QPSK, 16QAM and 64QAM are used.

WiFi 5

The Standard IEEE 802.11ac is referred as WiFi 5.
This WiFi standard is successor to IEEE 802.11n (i.e. WiFi 4).
This is the first wifi standard in which beamforming has been added to the MIMO and multi-user MIMO feature are also
introduced. WiFi-5 supports higher throughput due to addition of higher bandwidths ( upto 160 MHz ), multi-user MIMO,
higher number of spatial streams ( upto 8) and higher number of modulation schemes (256 QAM).
It operates on 5 GHz and supports legacy multi-carrier (OFDM) and single carrier (DSSS, CCK)
modulation schemes and baseband modulation types (BPSK, QPSK, 16QAM, 64 QAM, 256QAM)
Various channel Bandwidths are supported which include 20 MHz, 40 MHz, 80 MHz and 160 MHz.
WiFi-5 supports maximum data rate of 6.93 Gbps ( using 160MHz bandwidth, 8 spatial streams, MCS9, 256QAM, with short guard interval)
and coverage range of approx. 80 m with 3 antenna (about 10 meter more than wifi-4).
MIMO configuration up to 4 x 4 can be employed. It supports single user transmission as well as multi-user transmissions.

WLAN 802.11ac basics➤

WiFi 6

Image Courtesy: Ruckus Networks

The standard IEEE 802.11ax is referred as WiFi 6, which is called 6th generation of WiFi.
This WiFi standard is successor to IEEE 802.11ac (i.e. WiFi 5).
WiFi 6 offers higher speed and greater coverage range compare to legacy wifi networks wifi-5, wifi-4, wifi-3 etc.
WiFi-5 operates at 2.4 GHz and 5 GHz frequency bands.

OFDMA concept has been introduced in both uplink and downlink directions in wifi-6.
The other major features which have been introduced in the wifi-6 include MU-MIMO, beamforming, 1024-QAM,
longer size OFDM symbol, higher number of spatial streams (up to 8),
Uplink resource scheduling without any contention unlike 802.11ac etc.
The another unique feature is BSS coloring.
Due to its high efficiency performance, it is also known as HEW (High Efficiency WLAN).
802.11ax offers better efficiency, network capacity, performance and user experience at reduced
latency.
WLAN 802.11ax basics, advantages and disadvantages➤

Difference between wifi 6 and wifi 5, wifi 4, wifi 3, wifi 2, wifi 1

Following table compares various parameters viz. speed, coverage range etc. of various wifi versions and
mentions comparison between them.

WiFi Version
Features

WiFi 1
• IEEE 802.11b standard
• Speed: 1, 2, 5.5 and 11 Mbps
• Range: 38 meters (indoor), 140 meters (outdoor)

WiFi 2
• IEEE 802.11a standard
•  6, 9, 12, 18, 24, 36, 48 and 54 Mbps
•  35 meters (indoors) and 120 meters (outdoors)

WiFi 3
• IEEE 802.11g standard
•  Speed : Same as supported by wifi-1 and wifi-2
•  Range : Same as supported by wifi-1 and wifi-2

WiFi 4
• IEEE 802.11n standard
•  Speed : Maximum up to 150 Mbps
•  Range : 70 meters (indoor), 250 meters (outdoor)

WiFi 5
• IEEE 802.11ac standard
•  Speed : 433 Mbits/sec(with 80 MHz and 1 Spatial Streams(SS) ); 6.933 Gbits/sec (with 160MHz, 8 SS)
•  Range : 80 meters with 3 antennas

WiFi 6
• IEEE 802.11ax standard
•  Speed : 600.4 Mbits/sec (with 80 MHz, 1 SS); 9.6078 Gbits/sec (with 160 MHz, 8 SS)
•  Range : Better than wifi-5 routers as beamforming is supported in wifi-6 routers.

Difference between wifi 5 and wifi 6

Following table mentions difference between wifi-5 and wifi-6 with respect to physical layer parameters.

WiFi-6 and WiFi-5 related links

advantages and disadvantages of 802.11ac (WiFi 5)
advantages and disadvantages of 802.11ax (WiFi 6)
BSS coloring in 11ax
RU in 802.11ax
MU-OFDMA in 802.11ax
MU-MIMO in 802.11ax
Difference between 802.11ac Wave1 and Wave2
Difference between 802.11n,11ac,11ad
WLAN 802.11ac tutorial
WLAN 802.11ax tutorial
WLAN 802.11ac frame structure
WLAN 802.11ac Physical Layer
802.11ac MAC Layer
802.11ax basics
WLAN 11a,11b,11n,11ac frame structures

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Circuit Switching vs Packet switching
Packet Switching vs Message switching
FTP vs HTTP 
FTP vs SMTP 
FTP vs TFTP
hub Vs. switch 
TCP vs UDP 

RF and Wireless Terminologies

SATELLITE 
RF 
Antenna 
Avionics 
Wireless 
LiFi vs WiFi 
MiFi vs WiFi 
BPSK vs QPSK 
BJT vs FET 
PDH vs SDH 

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