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What do we know about the Arithmetic Sequences?

Arithmetic sequences

Arithmetic sequences are, you guessed it, ordered sequences of numbers, which involve the constant difference between adjacent numbers.

The most common example of arithmetic sequences are obviously the lists of consecutive integers. For instance,

{1, 2, 3, 4, 5, 6}, or, usually, any list of the form {x, x + 1, x + 2, x + 3…}

Arithmetic sequences are quite similar, but the difference between the terms can be any numbers. For instance,

{3, 7, 10, 13, 16}, {1/2, 5/2, 9/2, 13/2}, {2, 4, 6, 8, 10, 12, 14}, or, in general {x, x + d, x + 2d, x + 3d…}

Everything one needs to know about an arithmetic sequence can be summed up in three axiomatic chunks of information:

1. The 1st (or the last) term

2. The variation between the terms

3. The total number of terms

In most cases, the first two units of information are stated. If one is summing even integers, the difference between them is two and the first and last numbers must both be even regardless of the range given in the problem.

The third chunk of information is something we usually have to derive ourselves. A simple example is used below to show how to do this.

How many numbers are there between 7 and 15, that are inclusive?

Common sense dictates that we should get the difference between the two numbers, 15 – 7 = 8, and that will be the total number of you know, the numbers!

However, If we use the brute force approach of actually counting the integers we find that our shortcut approach was off by 1: {7, 8, 9, 10, 11, 12, 13, 14, 15}.

If you count the numbers, you’ll find that there are exactly 9 of them in the list. It may seem like we should just use the brute force approach each time we have to solve this kind of problem, but for larger sets this would quickly become impractical.

So, why is our shortcut approach incorrect? It’s pretty simple – by minusing the 7, we don’t take it in the set, and therefore our count was off by one.

To redeem from this, we add 1 to the difference. If x is lesser than y, the total numbers between x and y, inclusive would be

y – x + 1

We have to improvise on this formula if we want to use it to count an arithmetic sequence where the difference between numbers is greater than 1.

If d is the difference between the numbers  in a given sequence, and x is less than y, then the total numbers in the sequence are

(y – x)/d + 1

We have to be a little careful while applying this formula. If we’re asked to obtain the number of multiples of 3 that come between 2 and 34, our first term would not be 2, it would be 3, and our last term wouldn’t be 34, it’s 33 – the reason being that we’re only counting multiples of 3.

So, now that we understand how to count the total numbers, we can move on to finding the sum of those terms.

See Also: 12 Tenses in English Grammar with examples

First we would need a couple definitions

1. The average or arithmetic mean of a list of numbers happens to be the sum of the terms.

For instance, the mean of the first 5 positive even numbers (2, 4, 6, 8, and 10) is

(2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6

2. The median of a list of ordered numbers is the middle number in the list or the mean of the two middle terms.

That is, if an ordered list has 5 numbers, the 3rd number will be the median but if an ordered list has 6 numbers, the average of the third and fourth number will be the median.

Arithmetic sequences of different numbers/terms have this property that the median and mean of the ordered list are the same number.

Mean of arithmetic sequence = Median of arithmetic sequence

Moreover,  the mean of an arithmetic sequence is simply the average of the 1st and the last term, and since the median is equal to mean, the median also happens to be the average of the first and last term.

If this seems odd, know that there is a geometric interpretation of the mean as well. The mean of two numbers on a line of numbers is the midpoint.

For instance, the mean of 1 and 7 is 4, because the distance from 1 to 4 is the same as the distance between 4 and 7.

In an arithmetic sequence, the midpoint between the first and last number is precisely the median.

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