Asked by Jackie Rojas on Jul 22, 2024
Verified
Find the n th partial sum of the arithmetic sequence. 9,16,23,30,37,…,n=109,16,23,30,37 , \ldots , n = 109,16,23,30,37,…,n=10
A) 440
B) 790
C) 405
D) 395
E) 880
Arithmetic Sequence
A sequence of numbers in which the difference between consecutive numbers is constant.
Nth Partial Sum
The sum of the first n terms of a sequence, used to determine the cumulative value of a series up to a specified number of terms.
- Employ arithmetic progression formulas to identify certain terms and their corresponding sums.
Verified Answer
SS
Shelbee SheridanJul 26, 2024
Final Answer :
C
Explanation :
We can use the formula for the nth term of an arithmetic sequence:
an=a1+(n−1)da_n = a_1 + (n-1)dan=a1+(n−1)d
where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term we want to find.
For this sequence, we have:
an=9+(n−1)7a_n = 9 + (n-1)7an=9+(n−1)7
We want to find the 10th partial sum, which is the sum of the first 10 terms:
S10=9+16+23+…+a10=9+(9+7)+(9+2⋅7)+…+(9+9⋅7)=10⋅9+7⋅(1+2+…+9)=90+7⋅45=405\begin{align*}S_{10} &= 9 + 16 + 23 + \ldots + a_{10} \\&= 9 + (9+7) + (9+2\cdot7) + \ldots + (9+9\cdot7) \\&= 10\cdot9 + 7\cdot(1+2+\ldots+9) \\&= 90 + 7\cdot45 \\&= \boxed{405}\end{align*}S10=9+16+23+…+a10=9+(9+7)+(9+2⋅7)+…+(9+9⋅7)=10⋅9+7⋅(1+2+…+9)=90+7⋅45=405
Therefore, the best choice is (C).
an=a1+(n−1)da_n = a_1 + (n-1)dan=a1+(n−1)d
where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term we want to find.
For this sequence, we have:
an=9+(n−1)7a_n = 9 + (n-1)7an=9+(n−1)7
We want to find the 10th partial sum, which is the sum of the first 10 terms:
S10=9+16+23+…+a10=9+(9+7)+(9+2⋅7)+…+(9+9⋅7)=10⋅9+7⋅(1+2+…+9)=90+7⋅45=405\begin{align*}S_{10} &= 9 + 16 + 23 + \ldots + a_{10} \\&= 9 + (9+7) + (9+2\cdot7) + \ldots + (9+9\cdot7) \\&= 10\cdot9 + 7\cdot(1+2+\ldots+9) \\&= 90 + 7\cdot45 \\&= \boxed{405}\end{align*}S10=9+16+23+…+a10=9+(9+7)+(9+2⋅7)+…+(9+9⋅7)=10⋅9+7⋅(1+2+…+9)=90+7⋅45=405
Therefore, the best choice is (C).
Learning Objectives
- Employ arithmetic progression formulas to identify certain terms and their corresponding sums.