Q111 / 180 · COMEDK 2024 (Memory Based) · Mathematics
If the sum of $4^{\text {th }}$ and $7^{\text {th }}$ term of an AP is 31 and sum of first two term is 7 then find the $10^{\text {th }}$ term of $A P$ is
- A20
- Correct answerB29
- C33
- D28
Solution
Let $a$ be the first term and $d$ is the common difference also $a_n$ be the $n^{\text {th }}$ term.
$\begin{array}{ll}<br/>\therefore & a_4+a_7=31 \\<br/>& a+3 d+a+6 d=31 \quad\left[\because a_n=a+(n-1) d\right] \\<br/>& 2 a+9 d=31 ...(i)\\<br/>\therefore & a_1+a_2=7 \\<br/>& a+a+d=7 \\<br/>& 2 a+d=7 .....(ii)<br/>\end{array}$
Subtract Eq. (ii) from Eq. (i), we get
$\begin{aligned}<br/>& \Rightarrow \quad 2 a+9 d-2 a-d=31-7 \\<br/>& 8 d=24 \\<br/>& d=3 \\<br/>& \therefore \quad 2 a+3=7 \\<br/>& a=2 \\<br/>& \therefore \quad a_{10}=a+9 d \\<br/>& =2+27=29<br/>\end{aligned}$
$\begin{array}{ll}<br/>\therefore & a_4+a_7=31 \\<br/>& a+3 d+a+6 d=31 \quad\left[\because a_n=a+(n-1) d\right] \\<br/>& 2 a+9 d=31 ...(i)\\<br/>\therefore & a_1+a_2=7 \\<br/>& a+a+d=7 \\<br/>& 2 a+d=7 .....(ii)<br/>\end{array}$
Subtract Eq. (ii) from Eq. (i), we get
$\begin{aligned}<br/>& \Rightarrow \quad 2 a+9 d-2 a-d=31-7 \\<br/>& 8 d=24 \\<br/>& d=3 \\<br/>& \therefore \quad 2 a+3=7 \\<br/>& a=2 \\<br/>& \therefore \quad a_{10}=a+9 d \\<br/>& =2+27=29<br/>\end{aligned}$
Hence, the correct answer is option: 'B'.