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- Question 3 and 4 Exercise 4.1 @math-11-kpk:sol:unit04
- s of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... =Question 3(i)==== Write down the nth term of the sequence as suggested by the pattern. $\dfrac{1}{2}, \dfra... \ldots$ ====Solution==== We can reform the given sequence to pick the pattern of the sequence as: $$\dfrac{1}{1+1}, \dfrac{2}{2+1}, \dfrac{3}{3+1}, \dfrac{4}{4+1},..
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- s of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... 4,6,8, \ldots ,50$ ====Solution==== It is finite sequence whose last term is $50 $. =====Question 1(ii)===... 0,1,0,1, \ldots$. ====Solution==== It is infinite sequence, the last term may be $0$ or $1$ , but how much terms in this sequence, we don't know. =====Question 1(iii)===== Classi
- Unit 04: Sequence and Series (Solutions) @math-11-kpk:sol
- ===== Unit 04: Sequence and Series (Solutions) ===== This is a forth unit of the book Mathematics 11 publis... is unit the students will be able to * Define a sequence (progression) and its terms. * Know that a sequence can be constructed from a formula or an inductive def... al and pascal sequences. * Define an arithmetic sequence. * Find the nth or general term of an arithmeti
- Question 6 Exercise 4.1 @math-11-kpk:sol:unit04
- lutions of Question 6 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... ldots.$$ =====Question 6(i)===== Find the Pascal sequence for $n=5$ by using its general recursive definiti... GOOD ====Solution==== For $n=5$, we have Pascal sequence as follows: $$P_0=1, P_{r+1}=\dfrac{5-r}{r+1} P_r... nd{align} So, $0=P_7=P_8=...$. Hence the Pascal sequence for $n=5$ is $1,5,10,10,5,1,0,0,0, \ldots$. ====
- Question 1 Exercise 4.4 @math-11-kpk:sol:unit04
- lutions of Question 1 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... ===== Write the first five terms of geometric ric sequence given that $a_1=5, \quad r=3$ ====Solution==== The gcometric sequence is $a_1, a_1 r, a_1 r^2, a_1 r^3, a_1 r^4, \ldots... ===== Write the first five terms of geometric ric sequence given that $a_1=8, \quad r=-\dfrac{1}{2}$ ====Sol
- Question 4 & 5 Exercise 4.4 @math-11-kpk:sol:unit04
- ons of Question 4 & 5 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... on 4===== How many terms are there in a geometric sequence in which the first and the last terms are 16 and ... dfrac{1}{2}$ ? ====Solution==== First term of the sequence $a_1=16$\\ Last term of the sequence $a_n=\dfrac{1}{64}$ Common ratio $r=\dfrac{1}{2}$\\ We have to find $n
- Question 6 & 7 Exercise 4.4 @math-11-kpk:sol:unit04
- ons of Question 6 & 7 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... w that the reciprocal of the terms of a geometric sequence also form a geometric sequence. ====Solution==== Let we are considering the standard geometric sequence with common ratio $r$ that is\\ $$a_1, a_1 r, a_1
- Question 5 and 6 Exercise 4.2 @math-11-kpk:sol:unit04
- s of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... ar, Pakistan. =====Question 5===== Show that the sequence $$\log a, \log (a b), \log \left(a b^2\right), \l... on==== We first find $n$th term. Each term of the sequence is $\log$ of some number. Each log contains $a$ b... $$a_n=\log (a b^{n-1}).$$ We show that the given sequence is A.P. Since \begin{align}a_n&=\log(a b^{n-1}).
- Question 1 and 2 Exercise 4.2 @math-11-kpk:sol:unit04
- s of Question 1 and 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... ion 1===== Find the $15$th term of the arithmetic sequence $2,5,8, \ldots$ GOOD ====Solution==== Here $a_1=2... 2=44 \end{align} Hence the 15th term of the given sequence is $44$. =====Question 2===== The first term of an arithmetic sequence is 8 and the 21st is 108. Find the 7th term. GOOD
- Question 11 & 12 Exercise 4.3 @math-11-kpk:sol:unit04
- s of Question 11 & 12 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... econd $a_3=80 \mathrm{ft}$ and so on.\\ Hence the sequence $16,48,80, \ldots \quad$ is an arithmetic sequence with $d=48-16=32$.\\ The total distance in six second is... d day}&=Rs. 3 \text{ and so on}\end{align} So the sequence formed $1,2,3,4, \ldots$ is an arithmetic sequenc
- Question 2 & 3 Exercise 4.4 @math-11-kpk:sol:unit04
- ons of Question 2 & 3 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... 2===== Suppose that the third term of a geometric sequence is $27$ and the fifth term is $243$. Find the first term and common ratio of the sequence. ====Solution==== Here $$a_3=27 \quad\text{and}\q... ion 3===== Find the seventh term of the geometric sequence that has $2$ and $-\sqrt{2}$ for its second and t
- Question 2 Exercise 4.5 @math-11-kpk:sol:unit04
- lutions of Question 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... onents $a_1, a_n, n_2 r$ and $S_n$ of a geometric sequence are given. Find the ones that are missing $a_1=1,... onents $a_1, a_n, n_2 r$ and $S_n$ of a geometric sequence are given. Find the ones that are missing $r=\dfr... onents $a_1, a_n, n_2 r$ and $S_n$ of a geometric sequence are given. Find the ones that are missing $r=-2,
- Question 3 and 4 Exercise 4.2 @math-11-kpk:sol:unit04
- s of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... 25$. GOOD =====Question 4===== The $n$th term of sequence is given by $a_n=2n+7$. Show that it is an arithm... -7=2\end{align} This gives every two terms of the sequence has same distance $2$, hence it is an A.P. Putt
- Question 1 Exercise 4.3 @math-11-kpk:sol:unit04
- lutions of Question 1 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... um of the indicated number of terms in arithmetic sequence: $9,7,5,3, \ldots$; 20th term; 20 terms. GOOD ===... e indicated number of terms in case of arithmetic sequence: $3, \dfrac{8}{3}, \dfrac{7}{3}, 2, \ldots$; 11th
- Question 9 Exercise 4.4 @math-11-kpk:sol:unit04
- lutions of Question 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... _2, G_3, G_4, G_5, \dfrac{81}{2}$ forms geometric sequence of 7 terms, with $a_7=\dfrac{81}{2}$ and $a_1=\df... _3, G_4, G_5, G_6,-\dfrac{7}{64}$ forms geometric sequence of $8$ terms, with\\ \begin{align}a_8&=-\dfrac{7}