====== Question 1 Review Exercise 5 ======
Solutions of Question 1 of Review Exercise 5 of Unit 05: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

===== Question 1 =====
 Chose the correct option.
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i. If $t_n=6 n+5$ then $t_{n+1}=$  
    * %%(a)%% $6 n-1$  
    * (b) $6 n+11$
    * %%(c)%% $6 n+6$
    * (d) $6 n-5$ \\ <btn type="link" collapse="a1">See Answer</btn><collapse id="a1" collapsed="true">(b): $6 n+11$</collapse>

ii. The sum to infinity of the series:
$1+\dfrac{2}{3}+\dfrac{6}{3^2}+\dfrac{10}{3^3}+\dfrac{14}{3^4}+\ldots$
    * %%(a)%% $6$  
    * (b) $2$
    * %%(c)%% $3$ 
    * (d) $4$ \\ <btn type="link" collapse="a2">See Answer</btn><collapse id="a2" collapsed="true">(c): $3$ </collapse>

iii. Sum the series:$1+2.2+3.2^2+\cdots+100.2^{\prime \prime}$
    * (a) $99.2^{100}$ 
    * (b) $100.2^{100}$
    * %%(c)%% $99.2^{100}+1$
    * %%(d)%% $1000.2^{100}$  \\ <btn type="link" collapse="a3">See Answer</btn><collapse id="a3" collapsed="true">(c): $99.2^{100}+1$</collapse>

iv. The $n^{t h}$ term of the series: $1.2+2.3+3.4+\ldots$ 
    * (a) $n^2-n$
    * %%(b)%% $n^2+n$
    * %%(c)%% $n^2$ 
    * (d) None of these \\ <btn type="link" collapse="a4">See Answer</btn><collapse id="a4" collapsed="true">(b): $n^2+n$ </collapse>

v. Sum of $n$ terms of the series whose $n^{t h}$ term is $1+2^n$   
    * (a) $n \cdot 2^{n-1}$ 
    * %%(b)%% $(n+1)+2^{n+1}$ 
    * %%(c)%% $n+2(2^n-1)$
    * (d) None of these \\ <btn type="link" collapse="a5">See Answer</btn><collapse id="a5" collapsed="true">(c): $n+2(2^n-1)$</collapse>

vi. Evaluate $\Sigma\left(3+2^r\right)$, where $r=1,2,3, \ldots, 10$ 
    * %%(a)%% $2051$  
    * (b) $2049$ 
    * %%(c)%% $2076$ 
    * (d) $1052$ \\ <btn type="link" collapse="a6">See Answer</btn><collapse id="a6" collapsed="true">(c): $2076$</collapse>

vii. What is the $n$ term of the series: $1+\dfrac{1+2}{2}+\dfrac{1+2+3}{3}+\ldots$ 
    * %%(a)%% $\dfrac{n+1}{2}$  
    * (b) $\dfrac{n(n+1)}{2}$
    * %%(c)%% $n^2-(n+1)$ 
    * (d) $\dfrac{(n+1)(2 n+3)}{2}$ \\ <btn type="link" collapse="a7">See Answer</btn><collapse id="a7" collapsed="true">(a): $\dfrac{n+1}{2}$</collapse>

viii. Sum of $n$ terms of the series $1^3+3^3+5^3+7^3+\ldots$
    * (a) $n^2(2 n^2-1)$  
    * (b) $2 n^3+3 n^2$
    * %%(c)%% $n^3(n-1)$
    * (d) $n^3+8 n+4$ \\ <btn type="link" collapse="a7">See Answer</btn><collapse id="a7" collapsed="true">(a): $n^2(2 n^2-1)$</collapse>


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