JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Sequences and Series

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Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P's having at least 3 terms and at most 33 terms is ______.

Answer: 53

Solution

For an A.P. with first term

a = 100

, last term

ℓ = 199

, and

n

terms, the common difference is:

d = \frac{ℓ - a}{n - 1} = \frac{99}{n - 1}

For

d

to be an integer,

(n - 1)

must be a divisor of 99. Since

99 = 3^{2} \times 11

, the divisors are: 1, 3, 9, 11, 33, 99.

For

3 \leq n \leq 33

, we need

2 \leq n - 1 \leq 32

. The valid divisors are: 3, 9, 11.

The corresponding common differences are:

$n - 1 = 3$ : $d = 33$
$n - 1 = 9$ : $d = 11$
$n - 1 = 11$ : $d = 9$

Sum of common differences:

33 + 11 + 9 = 53

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Integer answer

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