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- Question 14, Exercise 4.5 @math-11-nbf:sol:unit04
- stion 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... nd fractional notation for the infinite geometric series; $0.444...$ ** Solution. ** We can express the ... = 0.4+0.04+0.004+...$$ This is infinite geometric series with $a_1=0.4$, $r=\frac{0.04}{0.4}=0.1$.\\ Since $|r|=0.1 < 1$, this series has the sum: \begin{align*} S-\infty & = \frac{a_
- Question 17, 18 and 19, Exercise 4.3 @math-11-nbf:sol:unit04
- 8 and 19 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... =====Question 17===== Find sum of the arithmetic series. $6+12+18+\ldots+96$. ** Solution. ** Given arithmetic series: $$6+12+18+\ldots+96.$$ So, $a_{1}=6$, $d=12-6=6... 102\\ &=1224. \end{align} Hence the sum of given series is $1224$. =====Question 18===== Find sum of the
- Question 14, 15 and 16, Exercise 4.7 @math-11-nbf:sol:unit04
- 5 and 16 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... Question 14===== Find the sum to $n$ terms of the series whose $n$th term is $n+1$. ** Solution. ** Consider $T_n$ represents the $n$th term of series, then $$ T_{n} = n+1. $$ Taking summation \begi... frac{n(n+3)}{2} \end{align*} Thus, the sum of the series is $\sum_{n=1}^{\infty} T_{n}= \dfrac{n(n+3)}{2}$
- Question 23 and 24, Exercise 4.7 @math-11-nbf:sol:unit04
- 3 and 24 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... . =====Question 23===== Sum to $n$ terms of the series (arithmetico-geometric series): $$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots.$$ ** Solution. ** Given arithmetic-geometric series: $$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ld
- Question 25 and 26, Exercise 4.7 @math-11-nbf:sol:unit04
- 5 and 26 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... n. =====Question 25===== Sum to $n$ term 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
- Unit 04: Sequences and Seeries @math-11-nbf:sol
- etic means between them. * Define an arithmetic series and establish the formula to find the sum of $n$ terms of the series. * Show that sum of $n$ arithmatic means betwee... thmetic sequence, arithmetic means and arithmetic series. * Define a geometric sequence and its general ... eometric means between them. * Define geometric series and find the sum of $n$ terms of a geometric seri
- Question 1 and 2, Exercise 4.3 @math-11-nbf:sol:unit04
- 1 and 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... Pakistan. =====Question 1===== Find the sum of series: $4+7+10+13+16+19+22+25$ ** Solution. ** Given: $4+7+10+13+16+19+22+25$. As the given series is arithmetic series with $a_1=4$, $d=7-4=3$, $n=8$, so \begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \im
- Question 5 and 6, Exercise 4.3 @math-11-nbf:sol:unit04
- 5 and 6 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... Pakistan. =====Question 5===== Find the sum of series. $a_{1}=50$, $n=20$, $d=-4$. FIXME Statement is logically incorrect. Find the sum of arithmetic series with: $a_{1}=50$, $n=20$, $d=-4$. ** Solution. *... $d=-4$.\\ Let $S_n$ represents sum of arithmetic series. Then \begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]
- Question 3 and 4, Exercise 4.3 @math-11-nbf:sol:unit04
- 3 and 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... stan. =====Question 3===== Find the sum of each series. $a_{1}=5$, $a_{n}=100$, $n=200$ FIXME Statement... logically incorrect. Find the sum of arithmetic series with: $a_{1}=5$, $a_{n}=100$, $n=200$. ** Soluti... $n=200$.\\ Let $S_n$ represents sum of arithmetic series. Then \begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\
- Question 7 and 8, Exercise 4.3 @math-11-nbf:sol:unit04
- 7 and 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... Pakistan. =====Question 7===== Find the sum of series. $9+11+13+15+\cdots$ for $n=12$ ** Solution. ** Given series is arithmetic series with $a_1=9$, $d=11-9=2$, $n=12$.\\ Let $S_n$ represents sum of the arithmetic serie
- Question 17 and 18, Exercise 4.7 @math-11-nbf:sol:unit04
- 7 and 18 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... amabad, Pakistan. =====Question 17===== Sum the series up to $n$ term: $2^{2}+5^{2}+8^{2}+\ldots$ ** So... $. \\ Now consider this to make kth term of given series by just taking square. </callout> Consider $T_k... 2+3n-1\right) \end{align*} Thus, the sum of the series is $\sum\limits_{k=1}^{n} T_{k} = \frac{n}{2}\le
- Question 27 and 28, Exercise 4.7 @math-11-nbf:sol:unit04
- 7 and 28 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... =====Question 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: $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ld... }{3}$. The sum of infinite arithmetico-geometric series is given by $$ S_{\infty}=\frac{a}{1-r}+\frac{d
- Question 29 and 30, Exercise 4.7 @math-11-nbf:sol:unit04
- 9 and 30 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... =====Question 29===== Find sum to infinity of the series: $$1+4 x+7 x^{2}+10 x^{3}+\ldots$$ ** Solution. ** The given arithmetic-geometric series is:\\ \[ 1 + 4x + 7x^2 + 10x^3 + \ldots \] It c... . The sum of the infinite arithmetico-geometric series is given by:\\ \[ S_{\infty} = \frac{a}{1 - r} +
- Question 1 and 2, Exercise 4.8 @math-11-nbf:sol:unit04
- 1 and 2 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... ing the method of difference, find the sum of the series: $3+7+13+21+\ldots$ to $n$ term. ** Solution. **... \\ & =n^2+n+1 \end{align*} Thus, the kth term of series: $$ T_{k}=k^2++k+1 $$ Now taking sum, we get \beg... 3}(n^2+3n+5) \end{align*} Hence the sum of given series is $\dfrac{n}{3}(n^2+3n+5)$. GOOD m( =====Qu
- Question 7 and 8, Exercise 4.8 @math-11-nbf:sol:unit04
- 7 and 8 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics fo... ==Question 7===== Find the sum of $n$ term of the series: $$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\fra... +\dots$$ Let $T_k$ represents the kth term of the series. Then \begin{align*} T_k &=\frac{1}{(3k-2)(3k+1)... frac{n}{3n+1} \end{align*} Hence the sum of given series is $\dfrac{n}{3n+1}$. GOOD =====Question 8====