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Question Number 17767 by Joel577 last updated on 10/Jul/17

Σ_(n = 1) ^(30) (n^2 + 1) = (A) Σ_(n = 1) ^(15) (2n^2 + 30n + 224) (B) Σ_(n = 1) ^(15) (2n^2 + 30n + 225) (C) Σ_(n = 1) ^(15) (2n^2 + 30n + 226) (D) Σ_(n = 1) ^(15) (2n^2 + 30n + 227) (E) Σ_(n = 1) ^(15) (2n^2 + 30n + 228)

$$\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{30}} {\sum}}\left({n}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\: \\ $$$$\left(\mathrm{A}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{224}\right) \\ $$$$\left(\mathrm{B}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{225}\right) \\ $$$$\left(\mathrm{C}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{226}\right) \\ $$$$\left(\mathrm{D}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{227}\right) \\ $$$$\left(\mathrm{E}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{228}\right) \\ $$

Answered by Tinkutara last updated on 11/Jul/17

Σ_(n=1) ^(30) (n^2 + 1) = 1^2 + 1 + 2^2 + 1 + ... + 30^2 + 1 = ((30×31×61)/6) + 30 = 9485 Now option (D) = ((n(n + 1)(2n + 1))/3) + 15n(n + 1) + 227n = ((15×16×31)/3) + 15×15×16 + 227×15 = 9485 Hence answer is (D).

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{30}} {\sum}}\left({n}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\:\mathrm{1}^{\mathrm{2}} \:+\:\mathrm{1}\:+\:\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{1}\:+\:...\:+\:\mathrm{30}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$=\:\frac{\mathrm{30}×\mathrm{31}×\mathrm{61}}{\mathrm{6}}\:+\:\mathrm{30}\:=\:\mathrm{9485} \\ $$$$\mathrm{Now}\:\mathrm{option}\:\left(\mathrm{D}\right)\:=\:\frac{{n}\left({n}\:+\:\mathrm{1}\right)\left(\mathrm{2}{n}\:+\:\mathrm{1}\right)}{\mathrm{3}} \\ $$$$+\:\mathrm{15}{n}\left({n}\:+\:\mathrm{1}\right)\:+\:\mathrm{227}{n} \\ $$$$=\:\frac{\mathrm{15}×\mathrm{16}×\mathrm{31}}{\mathrm{3}}\:+\:\mathrm{15}×\mathrm{15}×\mathrm{16}\:+\:\mathrm{227}×\mathrm{15} \\ $$$$=\:\mathrm{9485} \\ $$$$\mathrm{Hence}\:\mathrm{answer}\:\mathrm{is}\:\left(\mathrm{D}\right). \\ $$

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