Question

1.3+2.4+3.5+...+n(n+1)=1/6n(n+1)(2n+7) prove by pmi

Answer 1

Let's => n = (k+1)
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THEN,

$= > 1.3 + 2.4 + 3.5 + .....n(n + 2) \\ \\ = > 1.3 + 2.4 + ......(k + 1).(k + 3) \\ \\ = > 1.3 + 2.4 + 3.5 + ..... + k.(k + 2) + (k + 1).(k + 3) \\ \\ = > \frac{k(k + 1)(2k + 7)}{6} + (k + 1).(k + 3) \\ \\ = > \frac{k + 1}{6} (2 {k}^{2} + 7k + 6k + 18) \\ \\ = > \frac{k + 1}{6} (2 {k}^{2} + 13k + 18) \\ \\ = > \frac{k + 1}{6} (2 {k}^{2} + 9k + 4k + 18) \\ \\ = > \frac{k + 1}{6} [k(2k + 9) + 2(2k + 9)] \\ \\ = > \frac{(k + 1)(k + 2)(2k + 9)}{6} \\ \\ = > \frac{(k + 1)(k + 2)(2k + 2 + 7)}{6} \\ \\ = > \frac{(k + 1)(k + 1 + 1)(2(k + 1) + 7)}{6} \\ \\ = > \frac{n.(n + 1).(2n + 7)}{6} \: \: \: \: ...[k + 1 = n]$
____________________"[◢PROVED]"

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_-_-_-_☆$BE \: \: \: \: BRAINLY$☆_-_-_-_

Answer 2

Step-by-step explanation:

have a good day ...this is it