Question

2.The sum of 5th and 9th terms of AP is 72 and sum of 7th and 12th term is 97 find A.P.​

Answer 1

Answer:

hope it will help you........

Answer 2

Step-by-step explanation:

$\bf\underline{\underline{\red{Given:-}}}$

• The sum of 5th and 9th terms of AP is 72.

• The sum of 7th and 12th terms of AP is 97.

$\bf\underline{\underline{\blue{To\:Find:-}}}$

• The Arithmetic Progression (AP)

$\bf\underline{\underline{\green{Solution:-}}}$

As we know that

The nth term of an AP is given by the formula:-

$\boxed{\sf \leadsto a_{n} = a + (n - 1)d}$

The 5th term:-

$\sf \implies a_{5} = a + (5- 1)d$

$\sf = a + 4d$

$\sf The \: 5th \: term = a + 4d$

The 9th term:-

$\sf a_{9} = a + (9- 1)d$

$\sf = a + 8d$

$\sf The \: 9th \: term = a + 8d$

$\sf Given, \: The \: sum\: of\: 5th \: and\: 9th\: terms \:of \: AP \: is \: 72$

$\sf \implies a_{5} + a_{9} = 72$

$\sf \implies a + 4d + a + 8d= 72$

$\sf \implies 2a + 12d= 72......(i)$

The 7th term:-

$\sf \implies a_{7} = a + (7- 1)d$

$\sf = a + 6d$

$\sf The \: 7th \: term = a + 6d$

The 12th term:-

$\sf a_{12} = a + (12- 1)d$

$\sf = a + 11d$

$\sf The \: 12th \: term = a + 11d$

$\sf Given, \: The \: sum\: of\: 7th \: and\: 12th\: terms \:of \: AP \: is \: 97$

$\sf \implies a_{7} + a_{12} = 97$

$\sf \implies a + 6d + a + 11d= 97$

$\sf \implies 2a + 17d= 97......(ii)$

$\underline{\sf Subtracting \: equation \: (i) \: from \: (ii)}$

$\sf \implies 2a + 17d-(2a+12d)= 97-72$

$\sf \implies 2a + 17d- 2a-12d = 25$

$\sf \implies 5d = 25$

$\sf \implies d = 5$

Substituting d = 5 in equation (i)

$\sf \implies 2a + 12d= 72$

$\sf \implies 2a + 12(5) = 72$

$\sf \implies 2a + 60= 72$

$\sf \implies 2a = 72 - 60$

$\sf \implies 2a = 12$

$\sf \implies a = 6$

$\rm The \: first \: term = a = 6$

$\rm The \: second \: term = a+d = 6+5 = 11$

$\rm The \:third\: term = a+2d = 6+2(5) = 16$

$\rm The \: fourth\: term = a+d = 6+3(5) = 21$

$\underline{\boxed{\purple{\rm \therefore The\: AP = 6\: , 11\: , 16\: , 21}}}$