Question

$\blue{ \Large{\boxed{ \tt{ \red{Question:- }}}}}$
Justify whether it is true to say that the following are the nth terms of an AP.
$\tt{(1) \:2n -3}$
$\tt{(2)\:{3n}^{2}+5}$
$\tt{(3)\:1+n+{n}^{2}}$

$\small\underline\purple{\underline{\sf{\maltese\: Required\::-}}}$
$\red\rightarrow\tt\green\{Uncopied\: Solutions}$
$\red\rightarrow\tt\blue{full\: explanation}$


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Answer 1

Answer:

$\text{(i) \: Yes, Here } a_{n} = 2n - 3$

$\textbf{So, Put} \: n = 1 \: a_{1} = 2(1) - 3 = - 1$

$\textbf{Put} \: n = 2 \: a_{2} = 2(2) - 3 = 1$

$\textbf{Put} \: n = 3 \: a_{3} = 2(3) - 3 = 3$

$\textbf{Put} \: n = 4 \: a_{4} = 2(4) - 3 = 5$

$\sf \colorbox{god} {List of numbers becomes \: - 1 ,1 ,3 ,..}$

Here,

$⇒a_{2} - a_{1} = 1 - ( - 1) = 1 + 1 = 2$

$⇒a_{3} - a_{2} = 3 - 1 = 2$

$⇒a_{4} - a_{3} = 5 - 3 = 2$

$★ \textbf{So, it terms an AP }$

Hence,

$2n - 3\sf \colorbox{god} {is the nth term of an AP }$

$\text{ (ii) No. Here, \: } a_{n} = {3n}^{2} + 5$

$\textbf{Put} \: n = 1,a_{1} = 3 {(1)}^{2} + 5 = 8$

$\textbf{Put} \: n = 2 \: a_{2} = 3 {(2)}^{2} + 5 = 3(4) + 5 = 17$

$\textbf{Put} \: n = 3 ,a_{3} = 3 {(3)}^{2} + 5 = 3(9) + 5 = 27 + 5 = 32$

‣List of the becomes number becomes 8, 17, 32.

Here,

$⇒a_{2} - a_{1} = 17 - 8 = 9$

$⇒a_{3} - a_{2} = 32 - 17 = 15$

$∴a_{2} - a_{1}≠a_{3} - a_{2}$

Since,

‣The Successive difference of the list is not constant

$★\sf \colorbox{god} {So it does not form an AP}$

$\text{ (iii) No. Here, \: } a_{n} = 1 + n + {n}^{2}$

$\textbf{put \: } n = 1 \: a_{1} = 1 + 1 + {(1)}^{2} = 3$

$\textbf{put \: }n = 2 ,a_{2} = 1 + 2 + {(2)}^{2} = 1 + 2 + 4 = 7$

$\textbf{put \: }n =3 ,a_{3} = 1 + 3 + {(3)}^{2} = 1 + 3 + 9 = 13$

$\text{List of number becomes 3,7,13}$

Here,

$⇒a_{2} - a_{1} = 7 - 3 = 4$

$⇒a_{3} - a_{2} = 13 - 7 = 6$

$∴a_{2} - a_{1}≠a_{3} - a_{2}$

Since,

‣The Successive difference of the list is not constant

$★\sf \colorbox{god} {So it does not form an AP}$

$\\ \\ \\ \\ \sf \colorbox{lightgreen} {\red★ANSWER ᵇʸɴᴀᴡᴀʙ}$

Answer 2

Answer:

hope it will help you

Step-by-step explanation:

plz give more than 50 thanks