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$An arithmetic sequence is defined as follows: {[a_(1)=-29],[a_(i)=a_(i-1)+2]:} Find the sum of the first 48 terms in the sequence.$

An arithmetic sequence is defined as follows: $\newline$ $\left\{\begin{array}{l} a_{1}=-29 \\ a_{i}=a_{i-1}+2 \end{array}\right.$ $\newline$ Find the sum of the first $48$ terms in the sequence.

Full solution

Q. An arithmetic sequence is defined as follows:

\newline

\left\{\begin{array}{l} a_{1}=-29 \\ a_{i}=a_{i-1}+2 \end{array}\right.

\newline

Find the sum of the first

48

terms in the sequence.

Identify first term and common difference: Identify the first term and the common difference of the arithmetic sequence. The first term $a_1$ is given as $-29$ , and the common difference $d$ is the amount added to each term to get the next term, which is $2$ .
Find the $48$ th term: Use the formula for the nth term of an arithmetic sequence to find the $48$ th term. $\newline$ The nth term $a_n$ of an arithmetic sequence can be found using the formula $a_n = a_1 + (n - 1)d$ . Let's find $a_{48}$ . $\newline$ $a_{48} = a_1 + (48 - 1) \times 2$ $\newline$ $a_{48} = -29 + 47 \times 2$ $\newline$ $a_{48} = -29 + 94$ $\newline$ $a_{$48$} = $65$
Calculate the sum of the first $48$ terms: Use the formula for the sum of the first $n$ terms of an arithmetic sequence.$\newline$The sum $S_n$ of the first $n$ terms of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2} \times (a_{1} + a_{n})$. Let's find $S_{48}$.$\newline$$S_{48} = \frac{48}{2} \times (a_{1} + a_{48})$$\newline$$S_{48} = 24 \times (-29 + 65)$$\newline$$S_{48} = 24 \times 36$$\newline$$S_{48} = 864$