Question 1071364: The first,third and sixth terms of an AP correspond to the first three consecutive terms of an increasing G.P.The first term of each progression is 16,the common difference is d and the common ratio of the G.P is r.
(I)Write two equations involving d and r
(ii)find the value of d and r
Find the sum of the first 20 terms
(I)the arithmetic progression
(ii)the geometric progression
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  is term number  of the AP
 is term number of the GP
(I) "The first, third and sixth terms of an AP correspond to the first three consecutive terms of an increasing G.P." translates into the equationa
 (1)
and
 (2).
(ii) Adding equation (1) times  plus equation (2) times  we get
Rearranging,
Dividing both sides of the equal sign by 
Factoring, we get
 ,
so the solutions are  and  .
would make all terms of the GP equal to  is the only reasonable answer.
Substituting into equation (1), we get
Find the sum of the first 20 terms:
(I) The sum of the first terms of an AP can be calculated as
 , where  is the first term.
With  ,
(ii) The sum of the first terms of a GP can be calculated as
 , where  is the first term.
With  ,
We could ask a calculator and get
There is no way to simplify that fraction, and
as a decimal there would be 15 digits after the decimal point,
but an approximate value ois  .
Just for fun, if you like factoring,
it is also 
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