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Asked by 20sanjaykoley20 | 19 Dec, 2023, 12:57: AM

Each term in above sum is written as

Each term is arithmetic progression ( 4+7+10+........... ) . Number of terms in arithmetic progression (A.P.)equals term number .

For example first term of given sum  has only one term of  A.P. , second term of given sum fas two terms of A.P etc..

Given sum is written as

inner sum is written as

Hence in above sum , first term is sum of natural numbers and second term is sum of squares

Above expression is simplified as

Hence we get , S29 = (1/2) ( 29 × 30 × 32)    and  S9 = (1/2) ( 9 × 10 × 12)

S29 - S9 = 60 × [ 232 - 9 ]

(1/60) [ S29 - S] = 223

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