Examples of Math Problems: Suppose Tn= a+(n-1)d is an AP with first term "a" and difference "d". Show that the sequence Un=2^(Tn) is a GP and find its first term and ratio.

Friday, April 4, 2014

Suppose Tn= a+(n-1)d is an AP with first term "a" and difference "d". Show that the sequence Un=2^(Tn) is a GP and find its first term and ratio.

U%5Bn%5D=2%5ET%5Bn%5D=2%5E%28a%2B%28n-1%29%2Ad%29

U%5Bn%5D=2%5Ea%2A2%5E%28%28n-1%29%2Ad%29=%282%5E%28a-d%29%29%2A2%5E%28n%2Ad%29%0D%0A

take integer k, then
U%5Bk%2B1%5D=2%5Ea%2A2%5E%28k%2Ad%29

U%5Bk%2B1%5D=2%5Ea%2A2%5E%28k%2Ad%29

U%5Bk%5D=2%5Ea%2A2%5E%28%28k-1%29%2Ad%29

the ratio is
U%5Bk%2B1%5D%2FU%5Bk%5D=2%5E%28%28a%2Bkd%29-%28a-d%2Bkd%29%29=2%5Ed

U%5Bk%2B1%5D%2FU%5Bk%5D=2%5E%28%28a%2Bkd%29-%28a-d%2Bkd%29%29=2%5Ed

therefore it is geometric with
a%5B0%5D=2%5E%28a-d%29

a%5B0%5D=2%5E%28a-d%29

and

r=2%5Ed

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