Tammy Todd

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. $-9+\left(-12\right)+\left(-15\right)$ + ... (to 10 terms)

likvau

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 ${a}_{n}={a}_{1}+\left(n-1\right)d$ For $n=7,$ ${a}_{7}-{a}_{1}+\left(7-1\right)d$
$21=5+6d$
$6d=21-5$
$6d=16$ $d=\frac{7}{3}$ Step 3 Therefore, the five arithmetic means between 5 and 21 are ${A}_{1}={a}_{1}+d=5+\frac{8}{3}=\frac{23}{3}$
${A}_{2}={a}_{1}+2d=5+2×\frac{8}{3}=\frac{31}{3}$
${A}_{3}={a}_{1}+3d=5+3×\frac{8}{3}=\frac{39}{3}=13$
${A}_{4}={a}_{1}+4d=5+4×\frac{8}{3}=\frac{47}{3}$
${A}_{5}={a}_{1}+5d=5+5×\frac{8}{3}=\frac{55}{3}$ 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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