Question 1 and 2, Exercise 4.6

Solutions of Question 1 and 2 of Exercise 4.6 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.

Find the indicated term of the harmonic progression. $\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \quad 7$ th term.

Solution.

$$\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \text{ is in H.P.}$$ $$9, 12, 15, ... \text{ is in A.P.}$$ Here $a_1=9$, $d=12-9=3$, $a_7=?$.

Gneral term of A.P is given as $$ a_n=a_1+(n-1)d. $$ Thus \begin{align*} a_7&=9+(6)(3) \\ & = 27 \text{ is in A.P} \end{align*} Hence the 7th term in H.P is $\dfrac{1}{27}$.

Find the indicated term of the harmonic progression. $\frac{1}{11}, \frac{1}{9}, \frac{1}{7}, \ldots \quad 10$ th term.

Solution.

\begin{align*} &\frac{1}{11}, \frac{1}{9}, \frac{1}{7}, \ldots\quad \text{ is in H.P.}\\ &11, 9, 7, \ldots \quad \text{ is in A.P.}\end{align*} Here $a_1 = 11$, $d = 9 - 11 = -2.$ $a_{10}=?$

The general term of the A.P. is given as $$ a_n = a_1 + (n-1)d. $$ Thus, \begin{align*} a_{10} &= 11 + (10-1)(-2) \\ &= 11 + 9(-2) \\ &= 11 - 18 \\ &= -7. \end{align*} Hence, the 10th term in H.P. is $-\frac{1}{7}.$

Question 3 & 4 >