Given-the-equality-1-3-5-2p-1-p-1-2-p-N-Show-this-equality-is-true-when-we-replace-p-by-p-1-

Question Number 116685 by mathocean1 last updated on 05/Oct/20

Given the equality :

1 + 3 + 5 + \dots + (2 p + 1) = {(p + 1)}^{2} p \in N^{*}

Show this equality is true when

we replace p by p + 1

Answered by TANMAY PANACEA last updated on 05/Oct/20

1 + (n - 1) 2 = 2 p + 1

2 n - 1 = 2 p + 1 \to n = p + 1

s u m o f (p + 1) t e r m s

\frac{p + 1}{2} [2 \times 1 + (p + 1 - 1) 2]

\frac{(p + 1)}{2} \times [2 p + 2] = {(p + 1)}^{2}

n o w s u m o f (p + 1 + 1) t e r m s f r o m L H S

\frac{(p + 2)}{2} [2 \times 1 + (p + 2 - 1) \times 2]

= \frac{(p + 2)}{2} \times [2 + 2 p + 2] \to {(p + 2)}^{2}

R H S {(p + 1 + 1)}^{2} \to {(p + 2)}^{2}

h e n c e p r o v e d