General term of sequence of real numbers

CAT 2021 Slot 3 · QA · Hard · Sequences & Series

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Consider a sequence of real numbers x1,x2,x3,…x_1, x_2, x_3, \dots such that xn+1=xn+n−1x_{n+1} = x_n + n - 1 for all n≥1n \ge 1. If x1=−1x_1 = -1 then x100=x_{100} =

  1. A.

    4949

  2. B.

    4849

  3. C.

    4850

  4. D.

    4950

Answer

C

Explanation

We have xn+1−xn=n−1x_{n+1} - x_n = n - 1.

Summing for n=1n = 1 to 9999: x100−x1=∑n=199(n−1)=0+1+2+⋯+98=98×992=4851x_{100} - x_1 = \sum_{n=1}^{99} (n - 1) = 0 + 1 + 2 + \dots + 98 = \frac{98 \times 99}{2} = 4851

Since x1=−1x_1 = -1: x100=−1+4851=4850x_{100} = -1 + 4851 = 4850.

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