If the sum of the series 54 + 51 + 48 + dots | vedclass

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Question

(A) The given series is an Arithmetic Progression $(AP)$ with first term $a = 54$ and common difference $d = 51 - 54 = -3$.Let the number of terms be

Vedclass · Accepted Answer

$18$

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(A) The given series is an Arithmetic Progression $(AP)$ with first term $a = 54$ and common difference $d = 51 - 54 = -3$.Let the number of terms be $n$. The sum of $n$ terms of an $AP$ is given by $S_n = \frac{n}{2} \{2a + (n - 1)d\}$.Given $S_n = 513$,we have:$513 = \frac{n}{2} \{2(54) + (n - 1)(-3)\}$$1026 = n(108 - 3n + 3)$$1026 = n(111 - 3n)$$1026 = 111n - 3n^2$$3n^2 - 111n + 1026 = 0$Dividing by $3$:$n^2 - 37n + 342 = 0$Factoring the quadratic equation:$(n - 18)(n - 19) = 0$Thus,$n = 18$ or $n = 19$.Checking $n = 19$: The $19^{th}$ term is $a_{19} = 54 + (19 - 1)(-3) = 54 - 54 = 0$.Since the sum of $18$ terms and $19$ terms are both $513$,and $18$ is the first value,$n = 18$ is the standard solution.

If the sum of the series $54 + 51 + 48 + \dots$ is $513$,then the number of terms is:

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