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a geometric sequence of first term a ,and last term l ,has n terms ,prove that product of its term p=(al)^n/2?
2 Answers
- Anonymous1 decade agoFavorite Answer
Suppose GP is a, ar, ar^2, ... ar^(n-1)
with l= ar^(n-1)
Product of terms
= a . ar . ar^2. ... . ar^(n-1)
= a^n . r ^ ( 1 +2 +3 + ....+(n-1))
Powers of r form an AP with sum = (n-1)n/2
Product = a^n . r ^(n.(n-1)/2)
= a ^ [(2)(n/2)] . r ^(n.(n-1)/2)
= {a ^(2) . r ^ (n-1) } ^ (n/2)
={a ^(1) .a ^(1) r ^ (n-1) } ^ (n/2)
={a ^(1) .{a ^(1) r ^ (n-1)} } ^ (n/2)
={a .l } ^ (n/2)
- 1 decade ago
The terms of a geometric sequence are of the following form:
a, ar, ar^2,..., a r^(n-1).
The product of those terms is (a^n)(r^[(n-1)n/2])
= ( a*[ar^(n-1)] )^(n/2) = (a*l)^(n/2)
