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Find the sum of the first 45 terms of the following series, to the nearest integer.

2,7,12,dots
Answer:

Find the sum of the first $45$ terms of the following series, to the nearest integer. $\newline$ $2,7,12, \ldots$ $\newline$ Answer:

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Full solution

Q. Find the sum of the first

45

terms of the following series, to the nearest integer.

\newline

2,7,12, \ldots

\newline

Answer:

Identify pattern: Identify the pattern in the series. $\newline$ The series starts at $2$ and each term increases by $5$ . This is an arithmetic series with a common difference ( $d$ ) of $5$ .
Find terms: Find the first term $a_1$ and the common difference $d$ of the series. $\newline$ The first term $a_1$ is $2$ and the common difference $d$ is $5$ .
Use formula: Use the formula for the sum of the first $n$ terms of an arithmetic series: $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$ . We need to find the sum of the first $45$ terms, so $n = 45$ .
Substitute values: Substitute the values into the formula. $S_{45} = \frac{45}{2} \times (2\times2 + (45 - 1)\times5)$
Perform calculations: Perform the calculations inside the parentheses first. $S_{45} = \frac{45}{2} \times (4 + 44\times5)$
Multiply $44$ : Multiply $44$ by $5$ . $\newline$ $S_{45} = \frac{45}{2} \times (4 + 220)$
Add $4$ : Add $4$ to $220$ . $\newline$ $S_{45} = \frac{45}{2} \times 224$
Multiply $224$ : Multiply $224$ by $45$ and then divide by $2$ . $\newline$ $S_{45} = \frac{224 \times 45}{2}$
Perform multiplication: Perform the multiplication. $S_{45} = \frac{10080}{2}$
Divide by $2$ : Divide $10080$ by $2$ to get the final sum. $\newline$ $S_{45} = 5040$

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