Question

How many terms of the AP 9 17 25 ....... must be taken to give a sum of 636 .​

Answer 1

★ Given :-

• AP = 9 , 17 , 25 ………

★ Have to Find :-

• How many terms of AP taken to give sum of 636$.$

★ Solution :-

First we have to know :-

➤ What is Sequence ?

• Order of numbers or terms in which they are arranged written by an.

➤ What is A.P ?

• A sequence is said to be an A.P if the difference of two consecutive terms is always same and this difference is called Common difference denoted by ' d ' and the first term of sequence is denoted by 'a'

Now, lets do the Question

From the given A.P we know

First term ↬ A1 = 9

Common Difference ↬ A2 - A1 = 17 - 9 = 8.

We know the formula of nth term of A.P

• a_n = a + ( n - 1 ) d

Sum of n terms when last term is given :-

• s_n = n/2 [ 2a + ( n - 1 ) d ]

_____________________________

Answer 2

$\sf \bf \huge {\boxed {\mathbb {QUESTION}}}$

$\tt How \:many \:terms\: of\: the \:AP \:9, 17, 25 .......\\\tt must \:be \:taken\: to\: give\: a \:sum\: of\: 636$

$\sf \bf \huge {\boxed {\mathbb {ANSWER}}}$

$\sf \bf {\boxed {\mathbb {GIVEN}}}$

• $\sf \bf a=9$
• $\sf \bf S_n=636$
• $\sf \bf d=8$

$\sf \bf {\boxed {\mathbb {TO\:PROVE}}}$

• $\tt Number \:of\:terms \:to\:get\:the\:sum=636$

$\sf \bf {\boxed {\mathbb {SOLUTION}}}$

$\tt FORMULA$

${\boxed {\boxed {\color{blue} {\sf \bf S_n=\dfrac{n}{2}[2a+(n-1)d]}}}}$

$\tt Substitute\: the \:values$

$\sf \bf \implies 636=\dfrac {n}{2}[2\times 9+(n-1)8]$

$\sf \bf \implies 636=\dfrac {n}{2}[18+8n-8]$

$\sf \bf \implies 636=\dfrac {n}{\cancel {2}}\times\cancel {2} [9+4n-4]$

$\sf \bf \implies 636=n[5+4n]$

$\sf \bf \implies 636=5n+4{n}^{2}$

$\sf \bf \implies 4{n}^{2}+5n-636=0$

$\tt It\:is\:in\:the \:form \:of\:{\boxed {\sf \bf a{x}^{2}+bx+c=0}}$

• $\sf \bf a=4$
• $\sf \bf b=5$
• $\sf \bf c=-636$

$\tt Formula$

${\boxed {\boxed {\color {green} {\sf \bf x = \dfrac{ - b± \sqrt{ {b}^{2} - 4ac} }{2a}}}}}$

$\tt Substitute\: the \:values$

$\sf \bf\implies n = \dfrac{ - 5± \sqrt{ {(5)}^{2} - 4(4)(-636)} }{2(4)}$

$\sf \bf\implies n = \dfrac{ - 5± \sqrt{ 25 +10176} }{8}$

$\sf \bf \implies n = \dfrac{ - 5± \sqrt{10201} }{8}$

$\sf \bf\implies n = \dfrac{ - 5± {101} }{8}$

$\sf \bf\implies n = \dfrac{ - 5+ {101} }{8}\:or\:n = \dfrac{ - 5- {101} }{8}$

$\sf \bf\implies n = \dfrac{ 96 }{8}\:or\:n = \dfrac{ - 106 }{8}$

$\sf \bf\implies n=12 \:or\:n=\dfrac {-53}{4}$

$\tt n\: cannot\: be\: negative$

$\implies {\boxed {\boxed {\boxed {\color{red} {\sf \bf n=12}}}}}$

$\therefore \tt 12\:terms \:of \:the \:given\:AP\:should \:be\:added\:to\:get\:636$

$\sf \bf \huge {\boxed {\mathbb {HOPE \:IT \:HELPS \:YOU}}}$

_________________________________________

$\sf \bf \huge {\boxed {\mathbb {EXTRA\:INFORMATION}}}$

$\tt {Formulas \:of\:AP}$

$\sf \bf a_n=a+(n-1)d$

$\sf \bf S_n=\dfrac{n}{2}[a+a_n]$

$\sf \bf S_n=\dfrac{n}{2}[2a+(n-1)d]$

${\mathbb{\colorbox {orange} {\boxed{\boxed{\boxed{\boxed{\boxed{\colorbox {lime} {\boxed{\boxed{\boxed{\boxed{\boxed{\colorbox {aqua} {@suraj5070}}}}}}}}}}}}}}}$