SOLUTION: 1) 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. Thanks!!

Algebra ->  Sequences-and-series -> SOLUTION: 1) 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. Thanks!!       Log On


   

Question 826461: 1) 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.
Thanks!!

Answer by tommyt3rd(5050) About Me  (Show Source):
You can put this solution on YOUR website!
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%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
therefore it is geometric with
a%5B0%5D=2%5E%28a-d%29 and r=2%5Ed