Struggling with proof by induction. Can someone help me please with the below?

Question

A sequence an is defined recursively as an+2 = −3an+1 − 2an, a0 = 1, a1 = −2.
Conjecture a formula for an and prove it by induction.

Answer 1

a_n+2 = -3a_n+1 - 2a_n ;  a₀=1, a₁=-2

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a₂ = -3(-2) - 2(1) = 4

a₃ = -3(4) - 2(-2) = -8

a₄ = -3(-8) - 2(4) = 16

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Conjecture:       a_n = (-1)ⁿ2ⁿ ,  n≥0

 

a₀= (-1)⁰ * 2⁰ = 1   

a₁ = (-1)¹ * 2¹ = -2

a₂ = (-1)² * 2² = 4    (the conjectured equation follows the same pattern we saw above)

 

Assume the conjecture is true for n=k:   a_k = -1^k * 2^k

Note that this is equivalent to the recursive formula  a_k+1= -2*a_k

 

Now let n=k+1, if we can show it to be true for n=k+1, we're done.

 

      a_k+1= (-1)^k+1 * 2^k+1 = (-1)(2)(-1^k * 2^k ) = -2(a_k )

 

Done.