513 = n/2 { 108+( n-1)(-3)} . Find n.

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Solution

here 18 and 19 are possible. I think you are checking the sum of an Arithmetic progression. Here the 18th term of this progression is 3 and 19 th term is 0.so sum upto 18th term and 19th term will be equal.That's why this happens.

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S_{n} = 54 + 51 + 48 + . . . . n

= 513

n

n

2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} = 2^{n + 10}

lim n \to \infty \frac{1. n + 2 (n - 1) + 3 (n - 2) + . . . + (n - 2) .3 + (n - 1) 2 + n .1}{n^{3}}

\frac{11 (n - 1)}{2}

n

(n + 1)! = 42 (n - 1)!

(n + 3)! = 110 (n + 1)!

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