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- Question 1 Exercise 5.1 @math-11-kpk:sol:unit05
- stion 1 of Exercise 5.1 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Gr... hawar, Pakistan. =====Question 1(i)===== Sum the series $1^2+3^2+5^2+7^2+\ldots$ up to $n$ terms. ====Solution==== We see that each term of the given series is square of the terms of the series $1+3+5+\ldots$ whose $n^{\text {th }}$ term is $2 n-1$. Therefore
- Question 1 Exercise 5.3 @math-11-kpk:sol:unit05
- stion 1 of Exercise 5.4 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Gr... istan. =====Question 1(i)==== Find the sum of the series $\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\ld... $ terms. ====Solution==== The general term of the series is: $$T_n=\dfrac{1}{n(n+1)}$$ Resolving $T_n$ int... =\dfrac{n}{n+1} \end{align} Hence the sum of the series is: $$S_n=\dfrac{n}{n+1}$$ =====Question 1(ii)==
- Question 7 Review Exercise @math-11-kpk:sol:unit05
- on 7 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Gr... tan. =====Question 7(i)===== Find the sum of the series: $1.2^2+3.3^2+5.4^2+\ldots$ to $n$ terms. ====Solution==== The given series if the product of corresponding terms of the two series $1,3,5, \ldots,(2 n-1)$ and $2^2, 3^2, 4^2, \ldot
- Question 2 & 3 Exercise 5.1 @math-11-kpk:sol:unit05
- n 2 & 3 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Gr... $1.2+2.3+3.4+\ldots+99.100$. Solution: The given series is the product of the corresponding terms of the series $1+2+3+\ldots+99$ and $2+3+4+\ldots+100$, whose $... n^{\text {th }}$ terms are $n(n+1)$ and the given series have 99 terms. Therefore, the $n^{\text {th }}$ t
- Question 10 Exercise 7.3 @math-11-kpk:sol:unit07
- war, Pakistan. Q10 Find the sum of the following series: (i) $1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\ldots$ Solution: The given series is binomial series. Let it be identical with the expansion of $(1+x)^n$ that is $$ \begin{aligned} & 1+n ... } x^3+\ldots \end{aligned} $$ Comparing both the series, we have $n x=-\frac{1}{4}$ (I) and $\frac{n(n-1)
- Unit 04: Sequence and Series (Solutions) @math-11-kpk:sol
- ===== Unit 04: Sequence and Series (Solutions) ===== This is a forth unit of the book Mathematics 11 publ... ans tEtween two numtErs. * Define an arithmetic series. * Establish the formula to find the sum to n terms of an arithmetic series. * Show that sum of $n$ arithmetic means betwee... * Solve real life problems involving arithmetic series. * Define a geometric sequence. * Find the $n
- Question 9 Review Exercise @math-11-kpk:sol:unit05
- on 9 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Gr... )===== Find the sum of the first $n$ terms of the series $3+7+13+21+31+\ldots$ ====Solution==== Using meth... od of differences to compute the sum of the given series. \begin{align} & a_2-a_1=7-3=4 \\ & a_3-a_2=13-7=... \ldots \\ & a_n-a_{n-1}=(n-1) \text { term of the series } \\ & 4,6,8, \ldots \end{align} Adding column wi
- Question 1 Exercise 4.5 @math-11-kpk:sol:unit04
- estion 1 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... ts+3.2^9$ ====Solution==== In the given geometric series: $a_1=3, \quad r=\dfrac{6}{3}=2$ and $a_n=3.2^9$.\\ We first find $n$ and then the sum of series.\\ We know that $$a_n=a_1 r^{n-1}$$,\\ \begin{ali... ac{1}{16}$ ====Solution==== In the give geometric series $$a_1=8, \quad r=\dfrac{4}{8}=\dfrac{1}{2} \quad$
- Question 4 Exercise 4.5 @math-11-kpk:sol:unit04
- estion 4 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... \ldots .(\mathrm{i})\end{align}\\ It is geometric series with $$a_1=0.8, \quad r=0.1$$\\ We can find the i... align} The serics in braces is infinite gcometric series with $a_1=0.63, \quad r=0.01<1$.\\ Therefore the ... }\\ The sequence in bracket is infinite geometric series with $a_1=0.15, r=0.01<1$.\\ Thus the sum of the
- Question 2 & 3 Exercise 5.2 @math-11-kpk:sol:unit05
- n 2 & 3 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Gr... th }}$ term of the following arithmetic-geometric series: $\dfrac{0}{1}+\dfrac{1}{2}+\dfrac{2}{4}+\dfrac{3... +\dfrac{5}{32}+\ldots$ ====Solution==== The given series is the product of the corresponding terms of the series: $0,1.2,3, \ldots$ and $1, \dfrac{1}{2}, \dfrac{1
- Question 1 Exercise 5.2 @math-11-kpk:sol:unit05
- stion 1 of Exercise 5.2 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Gr... =====Question 1(i)===== Sum up to $n$ terms the series $1.2+2.2^2+3.2^3+4.2^4+\ldots$. ====Solution==== ... =====Question 1(ii)===== Sum up to $n$ terms the series $1+4 x+7 x^2+10 x^3+\ldots.$ ====Solution==== Let... =====Question 1(iii)===== Sum up to $n$ terms the series $1+2 x+3 x^2+4 x^3+\ldots$. ====Solution==== Let
- Question 1 Review Exercise 5 @math-11-kpk:sol:unit05
- n+11$</collapse> ii. The sum to infinity of the series: $1+\dfrac{2}{3}+\dfrac{6}{3^2}+\dfrac{10}{3^3}+\... llapsed="true">(c): $3$ </collapse> iii. Sum the series:$1+2.2+3.2^2+\cdots+100.2^{\prime \prime}$ * ... 100}+1$</collapse> iv. The $n^{t h}$ term of the series: $1.2+2.3+3.4+\ldots$ * (a) $n^2-n$ * %%... : $n^2+n$ </collapse> v. Sum of $n$ terms of the series whose $n^{t h}$ term is $1+2^n$ * (a) $n \
- Question 2 & 3 Review Exercise @math-11-kpk:sol:unit05
- & 3 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Gr... Peshawar, Pakistan. =====Question 2===== Sum the series to $n$ terms $1.2+2.3+3.4+\ldots$ ====Solution===... n+2)}{3}\end{align} =====Question 3===== Sum the series: $1.3 .5+2.4 .6+3.5 .7+\ldots$ to $n$ terms. ====Solution==== In the given series each term is the product of corresponding terms o
- Unit 05: Miscellaneous Series (Solutions) @math-11-kpk:sol
- ===== Unit 05: Miscellaneous Series (Solutions) ===== This is a fifth unit of the book Mathematics 11 pub... natural numbers. * Define arithmetico-geometric series. * Find sum to n terms of the arithmetico-geometric series. * Define method of differences. * Use this method to find the sum of $n$ terms of the series whose differences of the consecutive terms are ei
- Question 5 Exercise 4.1 @math-11-kpk:sol:unit04
- estion 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... ===Question 5(i)===== Write each of the following series in expanded form, $\sum_{j=1}^6(2 j-3)$ ====Solut... ==Question 5(ii)===== Write each of the following series in expanded form, $\sum_{k=1}^5(-1)^k 2^{k-1}$ ==... =Question 5(iii)===== Write each of the following series in expanded form, $\sum_{j=1}^{\infty} \dfrac{1}{