Question

the sum of 5th and 7th terms of an A. P is 56 and the 13th term is 42. find the A. P.​

Answer 1

a5 + a7 = 56

a +4d + a+6d = 56

2a+10d =56

a + 5d = 28 -1

a13 =42

a+12d =42 -2

by subtracting eq 1 and 2

a+5d=28

a+12d =42

we get -7d = -14

d =2

a+5d = 28

a +5×2=28

a =28-10

a=18

AP = 18 , 18+2 , 18+2×2 , 18+ 3×2

=18 , 20 , 22 , 24

please follow me

Answer 2

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

$\tt The\: sum\: of\: 5th \:and\: 7th\: terms\: of\: an \:A. P \:is \:56 \\\tt and \:the \:13th\: term\: is\: 42.\: Find\: the\: A. P.$

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

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

• $\sf \bf 5a+7a=56$
• $\sf \bf 13a=42$

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

• $\sf \bf Find \:A.P$

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

$\sf \bf\implies 5a+7a=56$

$\sf \bf \implies a+4d+a+6d=56$

$\implies {\boxed {\sf \bf 2a+10d=56}} -(i)$

$\sf \bf \implies 13a=42$

$\sf \bf {Multiply\: a+12d=42\:with\:2}$

$\sf \bf \implies 2 (a+12d=42)$

$\implies{\boxed {\sf \bf 2a+24d=84}}-(ii)$

$\tt {\underbrace {Substract\:equation\:(i)\:from\:equation(ii)}}$

$\sf \bf \: \cancel {2a} +24d=84$

$\sf \bf \cancel {-2a}-10d=-56$

______________________

$\sf \bf 14d=28$

$\sf \bf \implies d=\dfrac{28}{14}$

$\implies {\boxed {\color {red} {\sf \bf d=2}}}$

$\tt \underline {Substitute\:d=2\:in\:equation\:(i)}$

$\sf \bf \implies 2a+10d=56$

$\sf \bf \implies 2a+10(2)=56$

$\sf \bf \implies 2a+20=56$

$\sf \bf \implies 2a=56-20$

$\sf \bf \implies 2a=36$

$\sf \bf \implies a=\dfrac{36}{2}$

$\implies {\boxed {\color {red} {\sf \bf a=18}}}$

$\tt {\underbrace{General \:form\:of\:A.P}}$

$\sf \bf a, a+d, a+2d,a+3d........$

$\tt \underline {Substitute\: the\: values}$

$\sf \bf \implies 18, 18+2, 18+2(2),18+3(2)........$

$\sf \bf\implies 18, 20, 18+4,18+6........$

$\implies{\boxed {\boxed{\color{purple} {\sf \bf 18, 20, 22,24........}}}}$

$\therefore\tt The\:A. P\:is\:18, 20, 22,24.........$

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

_________________________________________

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

$\tt {\underbrace {Formulas \:of\:A. P}}$

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

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

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

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