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JEE Main 202228 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 28 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Sequences and Series

JEE Main 2022 · 28 June · Evening Shift · Mathematics · Sequences and Series

If n arithmetic means are inserted between a and 100 such that the...

If

n

arithmetic means are inserted between a and

100

such that the ratio of the first mean to the last mean is

1: 7

and

a + n = 33

, then the value of

n

is

Options

A $21$
B $22$
C $23$
D $24$

Correct answer

C. $23$

Step-by-step solution

Let the A.P. be $a, {A}_{1}, {A}_{2} . . . {A}_{n}, 100$

Here, common difference, $d = \dfrac{100 - a}{n + 1}$

Given $\dfrac{{A}_{1}}{{A}_{n}} = \dfrac{1}{7} ⇒ \dfrac{a + d}{100 - d} = \dfrac{1}{7} . . . \left(i\right)$

Also $a + n = 33$

From options, when $n = 23$ , $a = 10\text{and}d = \dfrac{90}{24} = \dfrac{15}{4}$

from \left(i\right) $\dfrac{10 + \dfrac{15}{4}}{100 - \dfrac{15}{4}} = \dfrac{55}{385} = \dfrac{1}{7}$

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