1. ## Another AP/GP question

The sum of the first 100 terms of an arithmetic progression is 10000; the first, second and fifth terms of this progression are three consecutive terms of a geometric progression. Find the first term a and the non-zero common difference, d, of the arithmetic progression. (Answer: a = 1, d = 2)

I found:

2a + 99d = 200

After which, I'm stuck.

Thanks in advance if you could help with this question!

2. Sum of first 100 terms = $\displaystyle \frac{100}{2}(2a+(100-1)d)=10000$

gives $\displaystyle 2a + 99d = 200$

1st term = $\displaystyle a$

2nd term = $\displaystyle a+d$

5th term= $\displaystyle a+4d$

Three terms of a geometric so exists $\displaystyle r$ such that

$\displaystyle ar = a + d$ and $\displaystyle ar^2=a+4d$

So $\displaystyle ar^2 = ar + 3d$ by subing in first equation into second

Also $\displaystyle ar^2 = ar + rd$ by multiplying both sides of first equation by $\displaystyle r$

So $\displaystyle r=3$

So $\displaystyle 3a = a+d$ and $\displaystyle 2a = d$

Sub into equation you got $\displaystyle d + 99d = 100d = 200$

$\displaystyle d=2$

$\displaystyle a = 1$

3. Yay thanks glaysher now I get it!

4. Originally Posted by margaritas
Yay thanks glaysher now I get it!
No problem