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FP1 Summation Question Help (OCR)

Hello everyone. I've come across a particular question in the Summation chapter of the OCR Further Pure 1 textbook, and I was wondering if anyone could write out a solution to PART 2 for me as I have not been able to get anything equivalent to the answer in the back of the textbook (Part 1 is ok, Part 2 is the problem). The question is Question 7, from Exercise 2C, Page 35, I've copied it down in the images below.

Part 1:

Part 2:

The answer given in the back of the textbook:
CodeCogsEqn (2).gif

Thanks a lot (to whoever replies)! :smile:

Reply 1

atsruser

Note that

a) from part 1

\sum (ra^r - (r-1)a^{r-1}) = (a-1)\sum ra^{r-1} + \sum a^{r-1}

b) the LHS telescopes down to something trivial
c) the first term on the RHS is (almost) what you want to find
d) the second term on the RHS is the sum of a GP