====== 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 $4$.
====Go to ====
[[math-11-nbf:sol:unit04:ex4-4-p9|< Question 18 & 19]]
[[math-11-nbf:sol:unit04:ex4-4-p11|Question 22 & 23 >]]