Question 28 and 29, Exercise 4.4
Solutions of Question 28 and 29 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 28
A super ball dropped from the top of the tower (556 ft high) always rebounds three-fourths of the distance fallen. How far (up and down) will the ball have travelled when it hits the ground for the 6th time?
Solution.
Question 29
The teaching staff of high school informs its members of school cancellation by telephone. The principal calls 2 teachers, each of whom in turn calls 2 other teachers, and so on. In order to inform the entire staff, 6 rounds of calls are made. Counting the principal, find how many people are in staff at high school?
Solution.
We have given
first person $=a_1= 1$ principal
People in first round $= a_2 = 2$
People in 2nd round $= a_3 = 2(2)=4$
…
People in 6ht round $= a_7$.
Thus we have the series
$$
1+2+4+...+a_7
$$
We have to find sum of geometric series with $a_1=1$, $r=2$, $n=7$.
As we have
$$
S_n=\frac{a_1\left(1-r^n \right)}{1-r}, \quad r\neq 1.
$$
Thus
\begin{align*}
S_6&=\frac{(1)\left(1-2^7 \right)}{1-2} \\
&=\frac{1-128}{-2}\\
&=127
\end{align*}
Hence 127 people are in staff at high school. 
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