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Chapter 11: Problem 36

Find the sum of the first 25 terms of the arithmetic sequence: \(7,19,31,43, \dots\)

Short Answer

Expert verified
The sum of the first 25 terms of the arithmetic sequence is 3775.

Step by step solution

01

Identify the common difference, first term and the number of terms

From the sequence, the first term \(a_1 = 7\), the common difference \(d = 19 - 7 = 12\), and the number of terms \(n = 25\). We can see that the sequence is arithmetic because the difference between any two successive terms is constant.
02

Find the 25th term

Use the formula \(a_n = a_1 + (n - 1) * d\) to calculate the 25th term \(a_n\). So, \(a_n = 7 + (25 - 1) * 12 = 295\). Thus, the 25th term of the sequence is 295.
03

Calculate the sum of the first 25 terms

Now, use the sum formula for an arithmetic sequence: \(S_n = n/2 * (a_1 + a_n)\). Insert the values and calculate the sum: \(S_{25} = 25/2 * (7 + 295) = 25/2 * 302 = 3775\). Therefore, the sum of the first 25 terms is 3775.

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Most popular questions from this chapter

Explaining the Concepts What are mutually exclusive events? Give an example of two events that are mutually exclusive.

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{M M M, M M F, M F M, M F F, F M M FMF, FFM, FFF\)} - Find the probability of selecting a family with $$\text{at least one male child.}$$

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