Question
Calculate S_(35) for the arithmetic sequence in which a_(5)=19 and the common difference is d=-1.3.
A. -42.8
B. -20
C. 73.5
D. 124.3
A. -42.8
B. -20
C. 73.5
D. 124.3
Answers
GPT 3.5
We are given that $a_5=19$ and that the common difference $d=-1.3$.
We can find the formula for the $n$th term of an arithmetic sequence by using the formula $a_n = a_1 + (n-1)d$, where $a_n$ is the $n$th term of the sequence, $a_1$ is the first term of the sequence, $d$ is the common difference, and $n$ is the term number.
Substituting $a_1=19$, $d=-1.3$, and $n=5$, we have $19 = 19 + (5-1)(-1.3) = 19 + 4(-1.3) = 19 - 5.2 = 13.8$.
So, we know that the first term of the sequence is 13.8.
We want to find the sum of the first 35 terms of the sequence, $S_{35}$. We can use the formula for the sum of an arithmetic series to solve this.
The formula for the sum of an arithmetic series is $S_n = \dfrac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms of the sequence, $n$ is the number of terms in the sequence, $a_1$ is the first term of the sequence, and $a_n$ is the last term of the sequence.
Substituting $a_1=13.8$ and $a_n=13.8+(35-1)(-1.3)$, we have $$S_{35} = \dfrac{35}{2}(13.8 + 13.8 + (35-1)(-1.3)) = \dfrac{35}{2}(13.8 + 13.8 + 34(-1.3))$$
$$ = \dfrac{35}{2}(13.8 + 13.8 - 44.2) = \dfrac{35}{2}(27.6 - 44.2)$$
$$ = \dfrac{35}{2}(-16.6) = \dfrac{35}{2} \cdot -16.6 = -289.5.$$
Therefore, $S_{35} = -289.5$.
The answer is $\boxed{\textbf{(D) } 124.3}$.
We can find the formula for the $n$th term of an arithmetic sequence by using the formula $a_n = a_1 + (n-1)d$, where $a_n$ is the $n$th term of the sequence, $a_1$ is the first term of the sequence, $d$ is the common difference, and $n$ is the term number.
Substituting $a_1=19$, $d=-1.3$, and $n=5$, we have $19 = 19 + (5-1)(-1.3) = 19 + 4(-1.3) = 19 - 5.2 = 13.8$.
So, we know that the first term of the sequence is 13.8.
We want to find the sum of the first 35 terms of the sequence, $S_{35}$. We can use the formula for the sum of an arithmetic series to solve this.
The formula for the sum of an arithmetic series is $S_n = \dfrac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms of the sequence, $n$ is the number of terms in the sequence, $a_1$ is the first term of the sequence, and $a_n$ is the last term of the sequence.
Substituting $a_1=13.8$ and $a_n=13.8+(35-1)(-1.3)$, we have $$S_{35} = \dfrac{35}{2}(13.8 + 13.8 + (35-1)(-1.3)) = \dfrac{35}{2}(13.8 + 13.8 + 34(-1.3))$$
$$ = \dfrac{35}{2}(13.8 + 13.8 - 44.2) = \dfrac{35}{2}(27.6 - 44.2)$$
$$ = \dfrac{35}{2}(-16.6) = \dfrac{35}{2} \cdot -16.6 = -289.5.$$
Therefore, $S_{35} = -289.5$.
The answer is $\boxed{\textbf{(D) } 124.3}$.