CAT Quantitative Ability Questions | CAT Number System questions

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CAT/2021.3

Question . 7

Consider a sequence of real numbers x₁, x₂, x₃, .... such that x_n+1 = x_n + n - 1 for all n ≥ 1. If x₁ = -1 then x₁₀₀

A : 4949 B : 4950 C : 4849 D : 4850

View Answer

Explanatory Answer

Method of solving this CAT Quantitative Ability Question from Number System question

Correct Option: D

To find the value of x100 in the given sequence, we can observe the pattern in the recursive formula:

xn+1 = xn + n - 1

We start with the given initial values:
x1 = -1
x2 = x1 + (1 - 1) = -1 + 0 = -1
x3 = x2 + (2 - 1) = -1 + 1 = 0

By analyzing the pattern, we notice that:
x3 = x1 + (1 - 1) + (2 - 1)
x4 = x1 + (1 - 1) + (2 - 1) + (3 - 1)

In general, we can write:
x100 = x1 + (1 - 1) + (2 - 1) + (3 - 1) + ... + (99 - 1)

Simplifying the expression:
x100 = x1 + (1 + 2 + 3 + 4 + ... + 99) - 99

The sum of consecutive numbers from 1 to 99 can be calculated using the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term) = (99/2)(1 + 99) = 4950

Substituting the values:
x100 = -1 + 4950 - 99
x100 = -1 + 4851
x100 = 4850

Therefore, the value of x100 in the sequence is 4850.

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