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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)`

A

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

B

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

C

Statement -1 is true , Statement -2 is false

D

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

Text Solution

AI Generated Solution

To prove that \( a_n = 3^n - 2^n \) for the given recurrence relation \( a_1 = 1, a_2 = 5 \) and \( a_{n+2} = 5a_{n+1} - 6a_n \) for \( n \geq 1 \), we will use mathematical induction. ### Step 1: Base Case We first check the base cases for \( n = 1 \) and \( n = 2 \). - For \( n = 1 \): \[ a_1 = 3^1 - 2^1 = 3 - 2 = 1 ...
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Knowledge Check

  • If a_(1)=5 and a_(n)=1+sqrt(a_(n-1)), find a_(3) .

    A
    2.623
    B
    2.635
    C
    2.673
    D
    2.799

  • If a_(0) = 0.4 and a_(n+1) = 2|a_(n)|-1 , then a_(5) =

    A
    `-0.6`
    B
    `-0.2`
    C
    0.2
    D
    0.4

  • If a_(1)=3 and a_(n)=n+a_(n-1) , the sum of the first five term is

    A
    17
    B
    30
    C
    42
    D
    45

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