# Question 20 and 21, Exercise 4.4

Solutions of Question 20 and 21 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

## Question 20

Find the missing geometric means. $$3 , \_\_\_ , \_\_\_ , \_\_\_ , 48$$

** Solution. **

We have given $a_1=3$ and $a_5=48$.

Assume $r$ be common difference, then by general formula for nth term, we have $$ a_n=ar^{n-1}. $$ This gives \begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = 2^4 \\ \implies & r = 2. \end{align*} Thus \begin{align*} & a_2=a_1 r= (3)(2) = 6 \\ & a_3=a_1 r^2 = (3)(2)^2 = 12 \\ & a_4=a_1 r^3= (3)(2)^3=24. \end{align*} Hence $6$, $12$, $24$ are required geometric means.

**The good solution is as follows:**

We have given $a_1=3$ and $a_5=48$.

Assume $r$ be common difference, then by general formula for nth term, we have $$ a_n=ar^{n-1}. $$ This gives \begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = (\pm 2)^4 \\ \implies & r = \pm 2. \end{align*} Thus, if $a_1=3$ and $r=2$, then \begin{align*} & a_2=a_1 r= (3)(2) = 6 \\ & a_3=a_1 r^2 = (3)(2)^2 = 12 \\ & a_4=a_1 r^3= (3)(2)^3=24. \end{align*} If $a_1=3$ and $r=-2$, then \begin{align*} & a_2=a_1 r= (3)(-2) = -6 \\ & a_3=a_1 r^2 = (3)(-2)^2 = 12 \\ & a_4=a_1 r^3= (3)(-2)^3=-24. \end{align*} Hence $6$, $12$, $24$ or $-6$, $12$, $-24$ are required geometric means.

## Question 21

Find the missing geometric means. $$1 ,\_\_\_,\_\_\_, 8$$

** Solution. **

We have $a_1=1$ and $a_4=8$. Assume $r$ to be the common ratio. Then, by the general formula for the $n$th term, we have $a_n = a_1 r^{n-1}.$ This gives \begin{align*} a_4 &= a_1 r^3 \\ \implies 8 &= 1 \cdot r^3 \\ \implies r^3 &= 8 \\ \implies r^3 &= 2^3 \\ \implies r &= 2. \end{align*} Thus, we can find the missing terms: \begin{align*} a_2 &= a_1 r = 1 \cdot 2 = 2, \\ a_3 &= a_1 r^2 = 1 \cdot 2^2 = 4. \end{align*} Hence, the missing geometric means are $2$ and $4$.

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