First term of an A.P. of consecutive integers is {p^2} + 1 sum of (2p + 1) terms of this series can

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8. Sequences and Series

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The first term of an $A.P.$ of consecutive integers is ${p^2} + 1$ The sum of $(2p + 1)$ terms of this series can be expressed as

A

${(p + 1)^2}$

B

${(p + 1)^3}$

C

$(2p + 1){(p + 1)^2}$

D

${p^3} + {(p + 1)^3}$

Solution

(d) ${S_{2p + 1}} = \frac{{2p + 1}}{2}\{ 2({p^2} + 1) + (2p + 1 – 1)\,1\} $

$ = \left( {\frac{{2p + 1}}{2}} \right)\,(2{p^2} + 2p + 2) = (2p + 1)({p^2} + p + 1)$

$ = {p^3} + {(p + 1)^3}$.

Standard 11

Mathematics

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