Find the sum. 81 plus 76 plus 71 plus ... minus 364

Question

Not really understanding this. I get 200 less then the actual answer. is it not x(n)=a+d(n-1)?

Answer 1

Hi An:

Start by writing the question in math notation.

We need to find the sum, S_n = 81 + 76 + 71 + . . . + (-364) [ where n = the (unknown) number of terms

This is an arithmetic series with first term a= 81 and common difference d = -5

The sum of n terms of an arithmetic series is S_n = (n/2)(2a + (n - 1)d) . . . . . . . . . . . . . . eqn (1)

We know a and d, so all we need is the number of terms, n.

We find this from the equation for the n-th term, a_n = a + (n - 1)d

Using the last term, given above: -364 = 81 + (n - 1) (-5)

-445 = (n -1)(-5) [subtracting 81 from both sides

-445/(-5) = n - 1 [dividing both sides by -5

89 = n - 1

so, n = 90 [adding 1 to both sides

Now, we can find S₉₀ from eqn (1): S₉₀ = (90/2) (2 x 81 + 89 x (-5)) [plugging in values for a, d, and n

= 45 (162 - 445)

= 45 x (-283)

= -12,735

I believe your error was to confuse a_n, which is the n-th term, with S_n, which is the sum of the first n terms.