Insert five arithmetic means between 5 and 21. (Enter your answers from smallest to largest.) Find the sum of the first n terms of the arithmetic sequence. −9 + (−12) + (−15) + ... (to 10 terms)

Question
Polynomial arithmetic
asked 2021-01-31
Insert five arithmetic means between 5 and 21. (Enter your answers from smallest to largest.) Find the sum of the first n terms of the arithmetic sequence. \(−9 + (−12) + (−15)\) + ... (to 10 terms)

Answers (1)

2021-02-01
Step 1 Let \(A_1, A_2, A_3, A_4, A_5\) be five arithmetic means between 5 and 21. Then \(5, A_1, A_2, A_3, A_4, A_5, 21\) are in arithmetic sequence with first term and seventh term are \(a_1 = 5,\)
\(a_7 = 21\) Step 2 Now nth term of arithmetic sequence is \(a_n = a_1 + (n - 1)d\) For \(n=7,\) \(a_7 - a_1 + (7 - 1)d\)
\(21 = 5 + 6d\)
\(6d = 21 - 5\)
\(6d = 16\) \(d = \frac{7}{3}\) Step 3 Therefore, the five arithmetic means between 5 and 21 are \(A_1 = a_1 + d = 5 + \frac{8}{3} = \frac{23}{3}\)
\(A_2 = a_1 + 2d = 5 + 2 \times \frac{8}{3} = \frac{31}{3}\)
\(A_3 = a_1 + 3d = 5 + 3 \times \frac{8}{3} = \frac{39}{3} = 13\)
\(A_4 = a_1 + 4d = 5 + 4 \times \frac{8}{3} = \frac{47}{3}\)
\(A_5 = a_1 + 5d = 5 + 5 \times \frac{8}{3} = \frac{55}{3}\) Step 4 Ans:The five arithmetic means between 5 and 21 are \(\frac{23}{3}, \frac{31}{3}, 13, \frac{47}{3}, \frac{55}{3}\)
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