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I'll try to answer the question, "If the 5th term of a geometric progression is 2, then the product of its FIRST 9 terms is --?"
Given the first term is A and the ratio is r, then the progression starts out...
A, Ar, Ar^2, Ar^3, Ar^4, ...
So the 5th term is Ar^4, which equals 2.
The series continues...
Ar^5, Ar^6, Ar^7, Ar^8, ...
Ar^8 is the 9th term.
The product P of all 9 terms is therefore:
P = A * Ar * Ar^2 *...*Ar^8
Collect all the A's
P = (A^9)*(1 * r * r^2 ...* r^8)
P = A^9 * r^(0+1+2+...+8)
There's a formula for the sum of the first n integers (n/2)(n+1), or if you don't know just add it up.
1+2+...+8 = 36
Therefore
P = A^9 * r^36
Since 36 is a multiple of 9, you can simplify:
P = (Ar^4)^9
Still with me? Remember that Ar^4=2 (a given fact).
So finally
P = 2^9 = 512.
Cute problem.
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∙ 9y ago
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