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

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Question

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

Vedclass · Accepted Answer

$18$

Vedclass · Answer

(A) The given series is an Arithmetic Progression $(AP)$ where the first term $a = 54$ and the common difference $d = 51 - 54 = -3$.Let the number of terms be $n$. The sum of the first $n$ terms of an $AP$ is given by the formula $S_n = \frac{n}{2} [2a + (n - 1)d]$.Given $S_n = 513$, we have:$513 = \frac{n}{2} [2(54) + (n - 1)(-3)]$$513 = \frac{n}{2} [108 - 3n + 3]$$1026 = n(111 - 3n)$$1026 = 111n - 3n^2$$3n^2 - 111n + 1026 = 0$Dividing the equation by $3$, we get:$n^2 - 37n + 342 = 0$Factoring the quadratic equation:$(n - 18)(n - 19) = 0$Thus, $n = 18$ or $n = 19$.Checking $n = 19$: The $19^{th}$ term is $a_{19} = a + 18d = 54 + 18(-3) = 54 - 54 = 0$. Since the $19^{th}$ term is $0$, the sum of $18$ terms and $19$ terms is the same $(513)$. However, in standard mathematical problems of this type, we usually provide the smallest integer value or note both. Given the options, $18$ is the correct choice.

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

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