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Mathematics 17 Online
OpenStudy (anonymous):

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

OpenStudy (danjs):

have you done anything yet?

OpenStudy (anonymous):

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

OpenStudy (danjs):

first, you have to figure what an arithmetic progression is

OpenStudy (anonymous):

well I know what that is already

OpenStudy (danjs):

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

OpenStudy (danjs):

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

OpenStudy (anonymous):

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

OpenStudy (danjs):

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

OpenStudy (anonymous):

ok

OpenStudy (danjs):

do you know what a geometric series is?

OpenStudy (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

OpenStudy (danjs):

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

OpenStudy (anonymous):

Yes

OpenStudy (danjs):

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

OpenStudy (danjs):

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

OpenStudy (danjs):

exponential

OpenStudy (danjs):

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

OpenStudy (danjs):

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

OpenStudy (danjs):

\[\frac{ a+4d }{ a } = \frac{ a+14d }{ a+4d }\] = r

OpenStudy (danjs):

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

OpenStudy (anonymous):

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

OpenStudy (danjs):

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

OpenStudy (danjs):

cross multiply first, for example

OpenStudy (danjs):

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

OpenStudy (danjs):

expand all those out from disributing

OpenStudy (danjs):

should be left with what they are looking for

OpenStudy (anonymous):

Ok, will try that quickly

OpenStudy (anonymous):

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

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