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- Question 1 Exercise 4.5
- 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 $$a_1=8, \quad r=\dfrac{4}{8}=\dfrac{1}{2} \quad$
- Question 4 Exercise 4.5
- estion 4 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... \ldots .(\mathrm{i})\end{align}\\ It is geometric series with $$a_1=0.8, \quad r=0.1$$\\ We can find the i... align} The serics in braces is infinite gcometric series with $a_1=0.63, \quad r=0.01<1$.\\ Therefore the ... }\\ The sequence in bracket is infinite geometric series with $a_1=0.15, r=0.01<1$.\\ Thus the sum of the
- Question 5 Exercise 4.1
- estion 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... ===Question 5(i)===== Write each of the following series in expanded form, $\sum_{j=1}^6(2 j-3)$ ====Solut... ==Question 5(ii)===== Write each of the following series in expanded form, $\sum_{k=1}^5(-1)^k 2^{k-1}$ ==... =Question 5(iii)===== Write each of the following series in expanded form, $\sum_{j=1}^{\infty} \dfrac{1}{
- Question 7 & 8 Exercise 4.3
- on 7 & 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... $ terms. ====Solution==== We can reform the given series into three arithmetic series \begin{align}&(1+7+13+\ldots)+(3+9+15+\ldots)- \\ & (5+11+17+\ldots) \ldots \ldots \ldots . . .(1)\end{align} Now the series each one in parenthesis is arithmetic,\\ we can f
- Question 9 & 10 Exercise 4.5
- n 9 & 10 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... 9===== The sum of first six terms of a geometric series is $9$ times the sum of its three terms. Find the... == We know that the sum of $n$ terms of geometric series with common ratio $r$ and first term $a_1$ is:\\ ... r=3$. =====Question 10==== How many terms of the series: $1+\sqrt{3}+3+\ldots$ be added to get the sum $4
- Question 13 & 14 Exercise 4.5
- 13 & 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... Solution===== Adding 1 to both sides of the given series, we get $$1+y=1+\dfrac{x}{3}+\dfrac{x^2}{3^2}+\dfrac{x^3}{3^3}$$ Now the series on R.H.S of the above is geometric series with $a_1=1$, $r=\dfrac{x}{3}$, and $|r|=\dfrac{x}{3}<1$ becau
- Question 11 & 12 Exercise 4.5
- 11 & 12 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... =====Question 12===== Find an infinite geometric series whose sum is 6 and such that each term is four ti... t. ====Solution==== Let considering the geometric series with common ratio $r$ and first term $a_1$\\ $$a_
- Question 13 & 14 Exercise 4.3
- 13 & 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr... een $1$ and $50$ so that the sum of the resulting series will be $459$ . ====Solution==== We know that:\\
- Question 1 and 2 Exercise 4.1
- 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr
- Question 3 and 4 Exercise 4.1
- 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr
- Question 6 Exercise 4.1
- estion 6 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr
- Question 1 and 2 Exercise 4.2
- 1 and 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr
- Question 3 and 4 Exercise 4.2
- 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr
- Question 5 and 6 Exercise 4.2
- 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr
- Question 7 Exercise 4.2
- estion 7 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Gr