n^th term of given series is, #160; a _ n =n(n+1) ∴ S _ n =∑ _ k=1 ^ n a _ k =∑ _ k=1 ^ n k(k 1) =∑ _ k=1 ^ n k ^ 2 +∑ _ k=1 ^ n ...

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Byju's Answer

Standard XII

Mathematics

Greatest Binomial Coefficients

1× 2+2× 3+3× ...

Question

$1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + . . .$

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Solution

$n^{t h}$ term of given series is, $a_{n} = n (n + 1)$
$∴ S_{n} = \sum_{k = 1}^{n} a_{k} = \sum_{k = 1}^{n} k (k - 1) = \sum_{k = 1}^{n} k^{2} + \sum_{k = 1}^{n} k = \frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2} = \frac{n (n + 1)}{2} (\frac{2 n + 1}{3} + 1) = \frac{n (n + 1)}{2} (\frac{2 n + 4}{3}) = \frac{n (n + 1) (n + 2)}{3}$

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Similar questions

Find the mode from the given data.

1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2 , 3, 4, 1, 2, 3, 4, 1 ,2 , 3, 4, 5, 5, 1, 5, 5, 5, 3, 5, 1, 2, 4, 5, 6, 1, 2, 4, 2, 1

(1) 4² + 2⁴− 3²

(2) 3³ − 2²+ 2⁰

(3) 5³ − 2² + 4⁰

(4) 6¹ + 5⁰ − 4¹

(5) (−2)² + 1² − 2²

Q. Observe the following pattern

(1 \times 2) + (2 \times 3) = \frac{2 \times 3 \times 4}{3}

(1 \times 2) + (2 \times 3) + (3 \times 4) = \frac{3 \times 4 \times 5}{3}

(1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) = \frac{4 \times 5 \times 6}{3}

and find the value of
(1 × 2) + (2 × 3) + (3 × 4) + (4 × 5) + (5 × 6)

$\int \frac{1}{2 + 3 s i n x} d x$

Gopi spun the spinner 30 times and got the following results:

2, 1, 4, 2, 4, 5, 1, 4, 1, 5, 3, 1, 4, 2, 3, 1, 5, 4, 4, 2, 3, 3, 5, 4, 2, 2, 5, 3, 1, 4

What is the frequency of 4?

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1×2+2×3+3×4+4×5+...

nth term of given series is, an=n(n+1) ∴Sn=∑nk=1ak=∑nk=1k(k−1)=∑nk=1k2+∑nk=1k=n(n+1)(2n+1)6+n(n+1)2=n(n+1)2(2n+13+1)=n(n+1)2(2n+43)=n(n+1)(n+2)3

$1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + . . .$