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$An arithmetic sequence is defined as follows: {[a_(1)=89],[a_(i)=a_(i-1)-9]:} Find the sum of the first 33 terms in the sequence.$

An arithmetic sequence is defined as follows: $\newline$ $\left\{\begin{array}{l} a_{1}=89 \\ a_{i}=a_{i-1}-9 \end{array}\right.$ $\newline$ Find the sum of the first $33$ terms in the sequence.

Full solution

Q. An arithmetic sequence is defined as follows:

\newline

\left\{\begin{array}{l} a_{1}=89 \\ a_{i}=a_{i-1}-9 \end{array}\right.

\newline

Find the sum of the first

33

terms in the sequence.

Identify first term and common difference: Identify the first term and the common difference. $\newline$ The first term, $a_1$ , is given as $89$ . The common difference, $d$ , is the amount subtracted from each term to get the next term, which is $-9$ .
Use formula for sum of first n terms: Use the formula for the sum of the first $n$ terms of an arithmetic sequence. $\newline$ The sum of the first $n$ terms, $S_n$ , of an arithmetic sequence is given by the formula: $\newline$ $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$ $\newline$ where $a_1$ is the first term, $n$ is the number of terms, and $d$ is the common difference.
Plug in values into formula: Plug in the values for $a_1$ , $n$ , and $d$ into the formula. $\newline$ We have $a_1 = 89$ , $n = 33$ , and $d = -9$ . Plugging these values into the formula gives us: $\newline$ $S_{33} = \frac{33}{2} \times (2\times89 + (33 - 1)\times(-9))$
Simplify the expression: Simplify the expression. $\newline$ $S_{33} = \frac{33}{2} \times (178 + 32 \times (-9))$ $\newline$ $S_{33} = \frac{33}{2} \times (178 - 288)$ $\newline$ $S_{33} = \frac{33}{2} \times (-110)$
Calculate the sum: Calculate the sum. $\newline$ $S_{33} = \frac{33}{2} \times (-110)$ $\newline$ $S_{33} = 33 \times (-55)$ $\newline$ $S_{33} = -1815$