The sum of the common terms of the following three arithmetic progressions. 3, 7, 11, 15, .., 399, 2, 5, 8, 11, ..., 359 and 2, 7, 12, 17, ..., 197, is equal to ________ . [2023]

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Solution

Q.
The sum of the common terms of the following three arithmetic progressions. 3, 7, 11, 15, .., 399,

2, 5, 8, 11, ..., 359 and 2, 7, 12, 17, ..., 197, is equal to ________ . [2023]

Ans.
(321)

$3, 7, 11, 15, \dots, 399$ are in A.P.

$So, common difference d_{1} = 7 - 3 = 4$

$2, 5, 8, 11, \dots, 359 are in A.P.$

$Then, d_{2} = 5 - 2 = 3; 2, 7, 12, 17, \dots, 197 are in A.P.$

$d_{3} = 7 - 2 = 5$

$L.C.M. of d_{1}, d_{2}, d_{3} = L.C.M. of (4, 3, 5) = 60$

$On expanding all A.P.s, common terms are 47, 107, 167$

$∴$ $Sum = 47 + 107 + 167 = 321$

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