7=====
Find the sum of $n$ term of the series: $$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\ldots$$
** Solution. **
Given: $$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \t
of the series (arithmetico-geometric series): $1+\frac{4}{7}+\frac{7}{7^{2}}+\frac{10}{7^{3}}+\ldots$
** Solution. **
The given arithmetic-geometric series is:
\[
1 + \frac{4}{7} + \frac{7}{7^2} + \frac{10}{7^3} + \ldots
\
Evaluate the sum of the series: $\sum_{k=1}^{n} \frac{1}{k(k+2)}$
** Solution. **
Let $T_k$ represent... h term of the series. Then
\begin{align*}
T_k &= \frac{1}{k(k+2)}.
\end{align*}
Resolving it into partial fractions:
\begin{align*}
\frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2} \ldots (1)
\end{align*}
Multiplying both sides by $k(k+2)$, we
t four terms of each arithmetic sequence. $a_{1}=\frac{3}{4}, d=\frac{1}{4}$
** Solution. **
Given: $a_1=\frac{3}{4}$, $d=\frac{1}{4}$.\\
We have
$$a_n = a_1 + (n - 1)d.$$
Now
\begin{align*}
a_2&=\frac{3}{4}+(2-1)\
27=====
Find sum to infinity of the series: $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$.
** Solution. **
Given arithmetic-geometric series is:
$$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$
It can b
is infinite geometric series with $a_1=0.4$, $r=\frac{0.04}{0.4}=0.1$.\\
Since $|r|=0.1 < 1$, this series has the sum:
\begin{align*}
S-\infty & = \frac{a_1}{1-r} \\
& = \frac{0.4}{1.0.1} = \frac{0.4}{0.9} \\
& = \frac{4}{9}.
\end{align*}
Hence $S_{\infty} =\dfrac{4}{9}$.
=====
f the geometric series. $a_{1}=343, a_{4}=-1, r=-\frac{1}{7}$
** Solution. **
Given $a_{1}=343$, $a_{4}=-1$, $r=-\frac{1}{7}$\\
Let $S_n$ represents the sum of geometric series. Then
$$ S_n =\frac{a_1-a_n r}{1-r}, \quad r\neq 1.$$
Thus
\begin{align*}
S_4 & =\frac{343-(-1)\left(-\frac{1}{7}\right)}{1+\frac{1}{7}}
=====Question 12=====
Find four H.Ms. between $\dfrac{1}{3}$ and $\dfrac{1}{11}$.
** Solution. **
Let $H_1, H_2, H_3, H_4$ be four $H.Ms$ between $\dfrac{1}{3}$ and $\dfrac{1}{11}$.\\
Then $$\dfrac{1}{3},H_1, H_2, H_3, H_4, \dfrac{1}{11} \text{ are in H.P.}
estion 9=====
Evaluate the sum of the series:
$$\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\ldots \ldots \text{ up to } \infty$$
** Solution. **
Do yourself... Evaluate the sum of the series: $\sum_{k=3}^{n} \dfrac{1}{(k+1)(k+2)}$
** Solution. **
Consider
\begin