Multiple choice

In the arithmetic sequence the first term is $8$, the common difference is $4$, and the sum of the first $n$ terms is $2808$. Find the value of $n$.

  1. $31$
  2. $36$
  3. $78$
  4. $41$
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B Correct answer
Explanation

The sum of an arithmetic series is given by S = n/2 * (2a + (n-1)d). Plugging in a=8, d=4, and S=2808 gives 2808 = n/2 * (16 + (n-1)4), which simplifies to 2808 = n(8 + 2n - 2) = 2n^2 + 6n. Solving the quadratic 2n^2 + 6n - 2808 = 0 or n^2 + 3n - 1404 = 0 yields n=36.

AI explanation

Using the arithmetic progression sum formula S = n/2 * [2a + (n - 1)d], we substitute the given values: 2808 = n/2 * [2(8) + (n - 1)4]. Simplifying this yields 2808 = n(6 + 2n), which gives the quadratic equation 2n^2 + 6n - 2808 = 0. Dividing by 2 results in n^2 + 3n - 1404 = 0; factoring this gives (n + 37)(n - 36) = 0. Discarding the negative root leaves n = 36.

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