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How can I derive an equation to find the sum of an arithmetic sequence?

This isn't a requirement of many courses but understanding and proving mathematics has order is what makes mathematics unique and enjoyable to many.

Imagine the sum of a sequence with n terms, denoted Sn, which has an initial value, a, and a constant value, d, added on to each term.

For example, my sequence could be:

1, 6, 11, 16... which would make a = 1 and d = 5.

Then Sn = 1 + 6 + 11 + 16 +...

So to be more general,

Sn = + (a+d) + (a+2d) + ...+ (a+(n-2)d) + (a+(n-1)d)


Sn = (a+(n-1)d) + (a+(n-2)d) +...+ (a+2d) + a   [reverse Sn]

Add both the sums together: add the first term to the other first term, then the second to the other second and so on.

2Sn = (2a+(n-1)d) + (2a+(n-1)d) +...+ (2a+(n-1)d) + (2a+(n-1)d)

There are n amounts of (2a+(n-1)d), as there are n terms, so this can be factored out.

2Sn = n(2a + (n-1)d)

=> Sn = 0.5n(2a + (n-1)d)         [divide by 2]

Rhys H. GCSE Maths tutor, A Level Maths tutor, A Level Further Mathem...

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