Question

if a n + 1 is equal to n square + 2 n + 2 then an is equal to​

Answer 1

Step-by-step explanation:

Proof:

Basis Step: If n = 0, then LHS = 02 = 0, and RHS = 0 * (0 + 1)(2*0 + 1)/6 = 0 .

Hence LHS = RHS.

Induction: Assume that for an arbitrary natural number n,

12 + 22 + ... + n2 = n( n + 1 )( 2n + 1 )/6. -------- Induction Hypothesis

To prove this for n+1, first try to express LHS for n+1 in terms of LHS for n, and use the induction hypothesis.

Here let us try

LHS for n + 1 = 12 + 22 + ... + n2 + (n + 1)2 = ( 12 + 22 + ... + n2 ) + (n + 1)2

Using the induction hypothesis, the last expression can be rewritten as

n( n + 1 )( 2n + 1 )/6 + (n + 1)2

Factoring (n + 1)/6 out, we get

( n + 1 )( n( 2n + 1 ) + 6 ( n + 1 ) )/6

= ( n + 1 )( 2n2 + 7n + 6 )/6

= ( n + 1 )( n + 2 )( 2n + 3 )/6 ,

which is equal to the RHS for n+1.

Thus LHS = RHS for n+1.

Answer 2

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