Question

The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th term is 44. Find the first three terms of the AP.​

Answer 1

Answer:

$\boxed{\bf AP : - 13, - 8, - 3,... a_n}$

Step-by-step explanation:

Given :

• The sum of 4th and 8th term of an AP is 24.
• The sum of the 6th and 10th term is 44.

To find :

• The first three terms of the AP.

Solution:

We know that,

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

Thus,

• $\sf a_4 = a + 3d$
• $\sf a_8 = a + 7d$

According to the question,

$\longrightarrow \sf a_4 + a_8 = 24 \\\\\longrightarrow \sf a + 3d + a + 7d = 24 \\\\\longrightarrow \sf 2a + 10d = 24 \\\\\longrightarrow \sf 2(a+5d) = 24 \\\\\longrightarrow \sf a + 5d = \frac{24}{2} \\\\\longrightarrow \sf a + 5d = 12 \: \: \: \dots (i)$

Again, The sum of the 6th and 10th term is 44.

• $\sf a_6 = a + 5d$
• $\sf a_{10} = a + 9d$

$\longrightarrow \sf a_6 + a_{10} = 44 \\\\\longrightarrow \sf a + 5d + a + 9d = 44 \\\\\longrightarrow \sf 2a + 14d = 44\\\\\longrightarrow \sf 2(a+7d) = 44 \\\\\longrightarrow \sf a + 7d = \frac{44}{2}\\\\\longrightarrow \sf a + 7d = 22 \: \: \: \dots (ii)$

On subtracting equation (i) from (ii),

$\longrightarrow \sf a + 5d -(a+7d)= 12-22\\\\\longrightarrow \sf a + 5d - a - 7d = - 10 \\\\\longrightarrow \sf - 2d = - 10 \\\\\longrightarrow \sf d = \frac{-10}{-2} \\\\\boxed{\bf \therefore \: d = 5}$

Substituting the value of d in (i),

$\sf \longrightarrow a + 5d = 12 \\\\\longrightarrow \sf a = 12 - 5d \\\\\longrightarrow \sf a = 12 - 5(5)\\\\\longrightarrow \sf a = 12 - 25 \\\\\boxed{\bf \therefore \: a = - 13}$

The three terms of AP are ;

• $\sf a_1 = - 13$
• $\sf a_2 = - 13 + 5 = - 8$
• $\sf a_3 = - 13 + 10 = - 3$

$\large{\boxed{\red{\bf AP : - 13, - 8, - 3,.. a_n}}}$

Answer 2

Given

$\bold{Sum \ of \ the \ 4th \ and \ 8th \ term} = 24\\\\\implies a+3d+a+7d= 24\\\implies 2a+10d=24\\\implies a + 5d = 12 ---- (1)$

$\bold{Sum \ of \ the \ 6th \ and \ 10th \ term} = 44\\\implies a + 5d + a + 9d = 44\\\implies 2a + 14 d = 44\\\implies a + 7d = 22 ---(2)$

Solving, 1 and 2 we get,

$a +5d = 12\\\underline{a+7d=22}\\\underline{\underline{-2d = -10}}\\\implies d = 5$

$a + 25 = 12\\a = -13\\Second \ term = -13 + 5 = -8\\Third \ term = -13 + 10 = -3$

$\boxed{\boxed{\bold{Therefore, \ the \ 3 \ terms \ of \ the \ AP \ are \ -13, \ -8 \ and \ -3}}}}$