CAT 2023 Slot 2 QA Question 20

Multiple choice (+3 / −1) · Algebra · Progression & Series · Try it, then check the answer and solution below.

CAT 2023 Slot 2QAAlgebra • Progression & SeriesModerate

Let both the series

a_1, a_2, a_3, \dots

and

b_1, b_2, b_3 \dots

be in arithmetic progression such that the common differences of both the series are prime numbers. If

a_5 = b_9 , a_{19} = b_{19}

and

b_2 = 0 , then

a_{11}$ equals

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: A

Answer and solution

Answer: A) $79$

📌 Core Concept & Formula

For arithmetic progressions

a_n = a_1 + (n-1)d_a

and

b_n = b_1 + (n-1)d_b

with prime common differences

d_a, d_b

a_{19} - a_5 = 14 d_a, \quad b_{19} - b_9 = 10 d_b

🔢 Step-by-Step Solution

Equate Differences:

Since

a_5 = b_9

and

a_{19} = b_{19}

a_{19} - a_5 = b_{19} - b_9 \implies 14 d_a = 10 d_b \implies 7 d_a = 5 d_b

Identify Prime Common Differences:

Since

d_a

and

d_b

are prime numbers, we must have:

d_a = 5, \quad d_b = 7

Determine the General Term of

b_n

Given

b_2 = 0

b_n = (n - 2) d_b = 7(n - 2) b_{19} = 7(19 - 2) = 7 \times 17 = 119

Calculate

a_{11}

Since

a_{19} = b_{19} = 119

a_{11} = a_{19} - 8 d_a = 119 - 8(5) = 119 - 40 = 79

⚡ 30-Second Shortcut

7 d_a = 5 d_b \implies d_a = 5, d_b = 7

.

b_{19} = 17 × 7 = 119$.

a_{11} = 119 - 8(5) = 79$.

🎯 Final Answer

Option A

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