The sum of the first 20 terms common between | vedclass

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Question

(B) The first series is $3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, \dots$ with common difference $d_1 = 4$.The second series is $1, 6, 11, 16,

Vedclass · Accepted Answer

$4020$

Vedclass · Answer

(B) The first series is $3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, \dots$ with common difference $d_1 = 4$.The second series is $1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, \dots$ with common difference $d_2 = 5$.The first common term is $11$. The next common term will be $11 + 	ext{lcm}(4, 5) = 11 + 20 = 31$.Thus,the common terms form an Arithmetic Progression ($A$.$P$.) with first term $a = 11$ and common difference $d = 20$.We need to find the sum of the first $n = 20$ terms of this $A$.$P$.The formula for the sum of $n$ terms is $S_n = \frac{n}{2} [2a + (n - 1)d]$.Substituting the values: $S_{20} = \frac{20}{2} [2(11) + (20 - 1)20]$.$S_{20} = 10 [22 + 19 	imes 20] = 10 [22 + 380] = 10 [402] = 4020$.Therefore,the sum is $4020$.

The sum of the first $20$ terms common between the series $3 + 7 + 11 + 15 + \dots$ and $1 + 6 + 11 + 16 + \dots$ is:

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