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Find the sum of the first 38 terms of the following series, to the nearest integer.

2,11,20,dots
Answer:

Find the sum of the first $38$ terms of the following series, to the nearest integer. $\newline$ $2,11,20, \ldots$ $\newline$ Answer:

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Geometric sequences

Full solution

Q. Find the sum of the first

38

terms of the following series, to the nearest integer.

\newline

2,11,20, \ldots

\newline

Answer:

Identify type and difference: Identify the type of series and the common difference. $\newline$ The series given is arithmetic because there is a constant difference between consecutive terms. $\newline$ To find the common difference, subtract the first term from the second term. $\newline$ $11 - 2 = 9$ $\newline$ Common Difference: $9$
Find $38$ th term: Find the $38$ th term of the series using the formula for the nth term of an arithmetic series. $\newline$ The nth term $a_n$ of an arithmetic series can be found using the formula: $\newline$ $a_n = a_1 + (n - 1)d$ $\newline$ where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number. $\newline$ For the $38$ th term: $\newline$ $a_{38} = 2 + (38 - 1) \times 9$ $\newline$ $a_{38} = 2 + 37 \times 9$ $\newline$ $a_{38} = 2 + 333$ $\newline$ $a_{38} = 335$
Calculate sum of terms: Calculate the sum of the first $38$ terms using the formula for the sum of an arithmetic series. $\newline$ The sum $S_n$ of the first $n$ terms of an arithmetic series can be found using the formula: $\newline$ $S_n = \frac{n}{2} \times (a_1 + a_n)$ $\newline$ where $a_1$ is the first term and $a_n$ is the nth term. $\newline$ For the sum of the first $38$ terms: $\newline$ $S_{38} = \frac{38}{2} \times (2 + 335)$ $\newline$ $S_{38} = 19 \times 337$ $\newline$ $S_{38} = 6403$

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