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
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)

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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