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$An arithmetic sequence is defined as follows: {[a_(1)=49],[a_(i)=a_(i-1)-3]:} Find the sum of the first 26 terms in the sequence.$

An arithmetic sequence is defined as follows: $\newline$ $\left\{\begin{array}{l} a_{1}=49 \\ a_{i}=a_{i-1}-3 \end{array}\right.$ $\newline$ Find the sum of the first $26$ terms in the sequence.

Full solution

Q. An arithmetic sequence is defined as follows:

\newline

\left\{\begin{array}{l} a_{1}=49 \\ a_{i}=a_{i-1}-3 \end{array}\right.

\newline

Find the sum of the first

26

terms in the sequence.

Identify Terms and Difference: Identify the first term and the common difference. $\newline$ The first term, $a_1$ , is given as $49$ . The common difference, $d$ , is the amount by which each term decreases, which is given as $-3$ (since each term is $3$ less than the previous one).
Use Sum Formula: 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 can be found using 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: Plug in the values for $a_1$ , $n$ , and $d$ into the sum formula. $\newline$ We have $a_1 = 49$ , $n = 26$ (since we want the sum of the first $26$ terms), and $d = -3$ . $\newline$ Now, we calculate the sum: $\newline$ $S_{26} = \frac{26}{2} \times (2\times49 + (26 - 1)\times(-3))$
Simplify Expression: Simplify the expression to find the sum. $\newline$ $S_{26} = 13 \times (98 + 25\times(-3))$ $\newline$ $S_{26} = 13 \times (98 - 75)$ $\newline$ $S_{26} = 13 \times 23$ $\newline$ $S_{26} = 299$