anonymous
  • 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
Mathematics
jamiebookeater
  • jamiebookeater
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DanJS
  • DanJS
have you done anything yet?
anonymous
  • anonymous
I actually don't even know where to start so no.
DanJS
  • DanJS
first, you have to figure what an arithmetic progression is

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anonymous
  • anonymous
well I know what that is already
DanJS
  • DanJS
Arithmetic sequence of numbers ---starts at a, and each consecutive term is 'd' away from the last term
DanJS
  • DanJS
a general form for the nth term is, \[a _{n} = a _{1} + (n - 1)*d\]
anonymous
  • anonymous
yes, i have the answers for the first question already figured out, its the second one i would like help with please
DanJS
  • DanJS
the nth term = the first term + (n-1) d
anonymous
  • anonymous
ok
DanJS
  • DanJS
do you know what a geometric series is?
DanJS
  • 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
DanJS
  • DanJS
The three terms from the last series to use are a ; a + 4d ; a + 14d
anonymous
  • anonymous
Yes
DanJS
  • DanJS
the common ratio between terms is r = (a + 4d) / a and r = (a + 14d)/(a+4d)
DanJS
  • DanJS
general geometric series to get the nth term Nth Term = (FIrst Term) * r^n
DanJS
  • DanJS
exponential
DanJS
  • DanJS
really all you have to do is know that the ratio of consecutive terms is always the same value r
DanJS
  • DanJS
set those two ratios equal, and solve for 3a to show it equals 8d
DanJS
  • DanJS
\[\frac{ a+4d }{ a } = \frac{ a+14d }{ a+4d }\] = r
DanJS
  • DanJS
solve for 3a = ... should come to that 8d value
anonymous
  • anonymous
Ok when you say solve for 3a how would I do that exactly (sorry)
DanJS
  • DanJS
just start moving things around and expanding and isolating a to one side and d to the other side
DanJS
  • DanJS
cross multiply first, for example
DanJS
  • DanJS
(a+4d)*(a+4d) = a*(a+14d)
DanJS
  • DanJS
expand all those out from disributing
DanJS
  • DanJS
should be left with what they are looking for
anonymous
  • anonymous
Ok, will try that quickly
anonymous
  • anonymous
Hooray I got the right answer! Thank you very much for your help!

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