Question

IF 9th TERM OF AN A.P. IS ZERO THEN PROVE THAT ITS 29th TERM IS DOUBLETHE 19th TERM

Answer 1

let 'a' and 'd' be the first term and common difference of an A.P

we have,
a9 = 0.

=> a = -8d --- (1)

now,
a29 = a + 28d
=> a29 = -8d + 28d [from (1)]
=> a29 = 20d --- (2)

and a19 = a+18d
=> a19 = -8d + 18d [from (1)]
=> a19 = 10d --- (3)
from (2) and (3)
we have
20d = 2×10d
=> a29 = 2×a19, proved

Answer 2

Question:-

➡ If the 9th term of an A.P. is zero then prove that, 29th term is twice the 19th term.

Proof:-

Let us assume that,

➡ First term of the A.P. = a and,

➡ Common Difference = d

Now,

Nth term of an A.P. = a + (n -1)d

So,

9th term = a + (9 - 1)d

= a + 8d

Now, it's given that, 9th term of the A.P. is zero.

➡ a + 8d = 0 .....(i)

Now,

29th term = a + (29 - 1)d

= a + 28d

19th term = a + (19 - 1)d

= a + 18d

Now,

29th term - 2 × 19th term

= a + 28d - 2 × (a + 18d)

= a + 28d - 2a - 36d

= -a - 8d

= -1(a + 8d)

= -1 × 0

= 0

Hence,

29th term - 2 × 19th term = 0

➡ 29th term = 2 × 19th term. (Hence Proved)