hi can someone help me find the sum of the first seven terms of the sequence when a1=7and r=2?

Answer 1

It sounds like you are dealing with a geometric sequence:

S = ar^0 + ar^1 + ar^2 + ... ar^n + ...

If so, then think about S(1 - r^k). It is S - Sr^k where:
Sr^k = ar^k +ar^(k+1) + ...

So S - Sr^k = ar^0 + ar^1 + ... + ar^(k-1)

For k = 1 we have: S - Sr = ar^0 = a
so S = a/(1-r)

Substituting for S in the above, we get:

S(1-r^k) = (a/(1-r))(1 - r^k) = a(1-r^k)/(1-r)

In this case, you want k = 7, r = 2, and a = 7.

(As it happens, when r >= 1, the summation doesn't converge so S is really undefined, but the formal manipulation still works and gets you the right answer here.

If you don't believe it, check the answer by doing division of polynomials:

(1 - x^2)/(1-x) = 1 + x
(1 - x^3)/(1-x) = 1 + x + x^2
...
(1 - x^7)/(1-x) = 1 + x + x^2 + ... + x^6