7 ; 3 ; -1 ; -5 ; ... Determine the sum of the first 50 terms of this sequence.

Question

Determine the sum of the first 50 terms of this sequence.

7 ; 3 ; -1 ; -5 ;...

Answer 1

This is an arithmetic sequence with a (first term)=7, common difference d=-4.

The sequence can be written a, a+d, a+2d, ..., a+(n-1)d where a+(n-1)d is the nth term.

We can rearrange this into pairs of terms:

(a+a+(n-1)d)+(a+d+a+(n-2)d)+...,

(2a+nd-d)+(2a+d+nd-2d)+...,

(2a+nd-d)+(2a+nd-d)+....

Because these pair-sums are identical and there are n/2 pairs, the sum of the series is:

S_n=(n/2)(2a+nd-d). Note that S₁=½(14+0)=7, the first term. So the formula works even for odd n.

If a=7 and d=-4, S₂=14-4=10 (n=2); S₂=7+3=10.

When n=4, S₄=2(14-12)=4; S₄=7+3-1-5=4.

So, for a=7, d=-4 and n=50, S₅₀=25(14+49×(-4))=25(14-196)=-4550.