How many terms of the A.P. 5, 8, 11, dots add | vedclass

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(C) For the given $A.P.$,the first term $a = 5$.The common difference $d = 8 - 5 = 3$ and the sum of $n$ terms $S_n = 670$.The formula for the sum of

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$20$

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(C) For the given $A.P.$,the first term $a = 5$.The common difference $d = 8 - 5 = 3$ and the sum of $n$ terms $S_n = 670$.The formula for the sum of $n$ terms is $S_n = \frac{n}{2} [2a + (n - 1)d]$.Substituting the values: $670 = \frac{n}{2} [2(5) + (n - 1)3]$.$1340 = n [10 + 3n - 3]$.$1340 = n [3n + 7]$.$3n^2 + 7n - 1340 = 0$.Factoring the quadratic equation: $3n^2 + 67n - 60n - 1340 = 0$.$n(3n + 67) - 20(3n + 67) = 0$.$(3n + 67)(n - 20) = 0$.Thus,$n = -\frac{67}{3}$ or $n = 20$.Since the number of terms $n$ must be a positive integer $(n \in N)$,$n = 20$ is the only valid solution.

How many terms of the $A.P.$ $5, 8, 11, \dots$ add up to $670$?

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