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arithmetic series help!
- Thread starter sammmmm
- Start date
Hello, sammmmm!
Do you know anything about arithmetic series?
\(\displaystyle \text{The }n^{th}\text{ term is: }\:a_n \:=\:a_1 + nd\)
\(\displaystyle \text{The sum of the first }n\text{ terms is: }\:S_n \:=\:\tfrac{n}{2}[2a_1 + (n-1)d]\)
. . \(\displaystyle \text{where: }\:\begin{Bmatrix} a_1 &=& \text{first term} \\ d &=& \text{common difference} \\ n &=& \text{number of terms} \end{Bmatrix}\)
\(\displaystyle \text{Second term is 10: }\:a_2 = 10 \quad\Rightarrow\quad a_1 + d \:=\:10 \;\;[1]\)
\(\displaystyle \text{Sum of first 18 terms is 1125: }\:S_{18} = 1125 \quad\Rightarrow\quad \tfrac{18}{2}(2a_1 + 17d) \:=1125 \quad\Rightarrow\quad 2a_1 + 17d \:=\:125 \;\;[2]\)
\(\displaystyle \text{We have a system of equations.}\)
. . \(\displaystyle \begin{array}{cccccc}\text{Multiply [1] by -2:} & \text{-}2a_1 - 2d &=& \text{-}20 \\ \text{Add [2]:} & 2a_1 + 17d &=& 125 \end{array}\)
\(\displaystyle \text{Therefore: }\:15d \:=\:105 \quad\Rightarrow\quad \boxed{d \:=\:7}\)
Do you know anything about arithmetic series?
\(\displaystyle \text{The }n^{th}\text{ term is: }\:a_n \:=\:a_1 + nd\)
\(\displaystyle \text{The sum of the first }n\text{ terms is: }\:S_n \:=\:\tfrac{n}{2}[2a_1 + (n-1)d]\)
. . \(\displaystyle \text{where: }\:\begin{Bmatrix} a_1 &=& \text{first term} \\ d &=& \text{common difference} \\ n &=& \text{number of terms} \end{Bmatrix}\)
The second term of an arithmetic series is 10, and the sum of the first 18 terms is 1125.
Find the common difference.
\(\displaystyle \text{Second term is 10: }\:a_2 = 10 \quad\Rightarrow\quad a_1 + d \:=\:10 \;\;[1]\)
\(\displaystyle \text{Sum of first 18 terms is 1125: }\:S_{18} = 1125 \quad\Rightarrow\quad \tfrac{18}{2}(2a_1 + 17d) \:=1125 \quad\Rightarrow\quad 2a_1 + 17d \:=\:125 \;\;[2]\)
\(\displaystyle \text{We have a system of equations.}\)
. . \(\displaystyle \begin{array}{cccccc}\text{Multiply [1] by -2:} & \text{-}2a_1 - 2d &=& \text{-}20 \\ \text{Add [2]:} & 2a_1 + 17d &=& 125 \end{array}\)
\(\displaystyle \text{Therefore: }\:15d \:=\:105 \quad\Rightarrow\quad \boxed{d \:=\:7}\)