Answer to Question #147323 in Discrete Mathematics for Manzar abbas

Question #147323
Use mathimatical Induction to prove that
1²+2²+3²+.......n²= n(n+1)(2n+1) divided by 6
1
Expert's answer
2020-12-01T02:11:12-0500

First we need to show the result is true for smallest value of n, in this case n = 0/

When n =0 the left side has only one term n2 = 0

The right side is

n(n+2)(2n+1)6=0(0+1)(0+1)6=0\frac{n(n+2)(2n+1)}{6} = \frac{0(0+1)(0+1)}{6} = 0

The second step is to show the statement is true for n = k, k+1 and so on.

Suppose, that statement is true for n = K.

12 + 22 + 32 + … + k2 = k(k+1)(2k+1)6\frac{k(k+1)(2k+1)}{6}

We need to prove, that statement is true fof n = k+1

12 + 22 + 32 + … + (k+1)2 = (k+1)(k+2)(2(k+1)+1)6\frac{(k+1)(k+2)(2(k+1) +1)}{6}

12 + 22 + 32 + … +k2 + (k+1)2 = (k+1)(k+2)(2(k+1)+1)6\frac{(k+1)(k+2)(2(k+1) +1)}{6}

We know, that

12 + 22 + 32 + … +k2 = k(k+1)(2k+1)6\frac{k(k+1)(2k+1)}{6}

So,

k(k+1)(2k+1)6+(k+1)2=k(k+1)(2k+1)+6(k+1)26\frac{k(k+1)(2k+1)}{6} + (k+1)^2 = \frac{k(k+1)(2k+1) +6(k+1)^2}{6}

=k+16(k(2k+1)+6(k+1))= \frac{k+1}{6}(k(2k+1) + 6(k+1))

=k+16(2k2+k+6k+6)= \frac{k+1}{6}(2k^2 + k + 6k + 6)

=k+16(2k2+7k+6)= \frac{k+1}{6}(2k^2 + 7k + 6)


2k2+7k+6=02k^2 + 7k + 6 = 0

2k2+4k+3k+6=02k^2 + 4k + 3k + 6 = 0

2k(k+2)+3(k+2)=02k(k+2) + 3(k+2) =0

(2k+3)(k+2)=0(2k+3)(k+2)=0


frack+16(2k2+7k+6)=(k+1)(k+2)(2k+3)6frac{k+1}{6}(2k^2 + 7k + 6) = \frac{(k+1)(k+2)(2k+3)}{6}

=(k+1)((k+1)+1)(2(k+1)+1)6= \frac{(k+1)((k+1)+1)(2(k+1)+1)}{6} = RHS hence proved


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