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

likvau

Skilled2021-02-01Added 75 answers

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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