## anonymous one year ago The 1st term of an arithmetic progression is a and the common difference is d, where d does not equal 0. (i) Write down expressions, in terms of a and d, for the 5th term and the 15th term. The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression. (ii) Show that 3a=8d I need help with (ii) please! I have the equations for (i) they are a + 4d ; a +14d

1. DanJS

have you done anything yet?

2. anonymous

I actually don't even know where to start so no.

3. DanJS

first, you have to figure what an arithmetic progression is

4. anonymous

well I know what that is already

5. DanJS

Arithmetic sequence of numbers ---starts at a, and each consecutive term is 'd' away from the last term

6. DanJS

a general form for the nth term is, $a _{n} = a _{1} + (n - 1)*d$

7. anonymous

yes, i have the answers for the first question already figured out, its the second one i would like help with please

8. DanJS

the nth term = the first term + (n-1) d

9. anonymous

ok

10. DanJS

do you know what a geometric series is?

11. DanJS

instead of an linear change from term to term, geometric series multiplies each by a common ratio r, every term / the one before it = r , and it is constant

12. DanJS

The three terms from the last series to use are a ; a + 4d ; a + 14d

13. anonymous

Yes

14. DanJS

the common ratio between terms is r = (a + 4d) / a and r = (a + 14d)/(a+4d)

15. DanJS

general geometric series to get the nth term Nth Term = (FIrst Term) * r^n

16. DanJS

exponential

17. DanJS

really all you have to do is know that the ratio of consecutive terms is always the same value r

18. DanJS

set those two ratios equal, and solve for 3a to show it equals 8d

19. DanJS

$\frac{ a+4d }{ a } = \frac{ a+14d }{ a+4d }$ = r

20. DanJS

solve for 3a = ... should come to that 8d value

21. anonymous

Ok when you say solve for 3a how would I do that exactly (sorry)

22. DanJS

just start moving things around and expanding and isolating a to one side and d to the other side

23. DanJS

cross multiply first, for example

24. DanJS

(a+4d)*(a+4d) = a*(a+14d)

25. DanJS

expand all those out from disributing

26. DanJS

should be left with what they are looking for

27. anonymous

Ok, will try that quickly

28. anonymous

Hooray I got the right answer! Thank you very much for your help!