If a(1)=1, a(2)=5 and a(n+2)=5a^(n+1)-6a...

If `a_(1)=1, a_(2)=5 and a_(n+2)=5a^(n+1)-6a_(n), n ge 1`. Show by using mathematical induction that `a_(n)=3^(n)-2^(n)`

Statement -1 is true , Statement -2 is true, Statement -2 is correct explanation for Statement -1

Statement -1 is true , Statement -2 is true , Statement -2 is not correct explanation for Staement -1

Statement -1 is true , Statement -2 is false

Statement -1 is false , Statement - 2 is true.

Text Solution

Verified by Experts

Let `P(n):a_(n)=3^(n)-2^(n)`.
Step I for `n=1`
LHS `=a_1=1`
and RHS `=3^1-2^1=1`
`therefore` LHS=RHS
Hence , P(1) is true .
For `n=2`
LHS `a_2=5`
and RHS `=3^2-2^2=5`
`therefore` LHS=RHS
Hence, P(2) is also true .
Thus ,P(1) and P(2) are ture .
Step II Let `P(k) and P(k-1)` are true
`therefore a_(k)=3^(k)-2^(k) and a_(k-1)=3^(k-1)-2^(k-1)`
Step III For `n=k+1`,
`a_(k+1)=5a_(k)-6a_(k-1)`
`=5(3^k-2^k)-6(3^(k-1)-2^(k-1))`
`=5.3^(k)-5.2^(k)-2.3^k+3.2^k`
`=3.3^k-2.2^k=3^(k+1)-2^(k+1)` which is true for `n=k+1`.
Hence, both statements are true and Statement-2 is a correct explanation of Statement -1.

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