The 4th term of an AP is thrice the first term and 7th term exceeds twice the 3th term by - Maths - Arithmetic Progressions

Question

The 4th term of an AP is thrice the first term and 7th term
exceeds twice the 3th term by 1 find the first term and C D

Lalit Mehra · Accepted Answer

Dear Student!

Let a and d be the first term and common difference of AP respectively.

n^thterm of A.P., a_n = a + (n – 1)d

Given, a₄ = 3a

∴ a + (4 – 1)d = 3a

$\Rightarrow a + 3 d = 3 a \Rightarrow 3 a - a = 3 d \Rightarrow 2 a = 3 d \Rightarrow a = \frac{3 d}{2} ... (1)$

Given, a₇ = 2 × a₃ + 1

$∴ a + (7 - 1) d = 2 [a + (3 - 1) d] + 1 \Rightarrow a + 6 d = 2 (a + 2 d) + 1 \Rightarrow a + 6 d = 2 a + 4 d + 1 \Rightarrow 6 d - 4 d = 2 a - a + 1 \Rightarrow 2 d = a + 1 ... (2)$

From 1( and (2), we get

$2 d = \frac{3 d}{2} + 1 ∴ 2 d - \frac{3 d}{2} = 1 \Rightarrow \frac{4 d - 3 d}{2} = 1 \Rightarrow \frac{d}{2} = 1 \Rightarrow d = 2$

When d = 2, we get

$a = \frac{3 \times 2}{2} = 3$ (Using(1))

Thus, the first term and common difference of A.P. is 3 and 2 respectively.

Cheers!

The 4th term of an AP is thrice the first term and 7th term exceeds twice the 3th term by 1. find the first term and C.D.