Find the number of terms in the arithmetic sequence with the given conditions. $a _ { 1 } = - 11$ , $d = \frac { 1 } { 4 } , S = - 195$
A) $n = 24$
B) $n = 30$
C) $n = 22$
D) $n = 48$
E) $n = 28$

Question

Quizplus · Accepted Answer

The sum of an arithmetic sequence is given by the formula:$$S_n = \frac{n}{2}(a_1 + a_n)$$where $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.We are given that $a_1 = -11$, $d = \frac{1}{4}$, and $S = -195$. We can use the formula for the sum of an arithmetic sequence to solve for $n$:$$-195 = \frac{n}{2}(-11 + a_n)$$$$-390 = n(-11 + a_n)$$$$a_n = -11 + \frac{390}{n}$$We can use the formula for the nth term of an arithmetic sequence to find $a_n$:$$a_n = a_1 + (n-1)d$$$$a_n = -11 + (n-1)\frac{1}{4}$$$$a_n = -11 + \frac{n}{4} - \frac{1}{4}$$$$a_n = -11 + \frac{n-1}{4}$$Setting the two expressions for $a_n$ equal to each other, we get:$$-11 + \frac{390}{n} = -11 + \frac{n-1}{4}$$$$\frac{390}{n} = \frac{n-1}{4}$$$$1560 = n^2 - n$$$$n^2 - n - 1560 = 0$$$$(n - 40)(n + 39) = 0$$$$n = 40, -39$$Since $n$ must be positive, we have $n = 40$. Therefore, the number of terms in the arithmetic sequence is 40.

Find the Number of Terms in the Arithmetic Sequence with the Given