Question

the 4th term of an A.P is three times the first and the 7th term exceeds twice the third term by 1 . find the first term and common difference​

Answer 1

Let a and be the first term and common difference of the A.P. respectively.

Since the 4th term is thrice the first term,

a₄=3a

⇒a+3d=3a ( Using:aₙ=a+(n-1)d )

⇒2a-3d=0 ...(1)

The 7th term exceeds twice the third term by 1.

⇒a₇=2a₃+1

⇒a+6d=2(a+2d) + 1 ( Using:aₙ=a+(n-1)d )

⇒a+6d=2a+4d+1

⇒a-2d= -1 ...(2)

Multiplying eq.(2) by 2,

2a-4d= -2 ...(3)

Subtracting eq.(2) from eq.(1),

2a-3d-2a+4d=0+2

⇒d=2

Putting d=2 in eq.(2),

a-2(2)= -1

⇒a-4= -1

⇒a=3

Hence, the first term and common difference of the A.P. are 3 and 2 respectively.