# If the 9th term of an AP is zero, then prove that its 29th term is twice its 19th term.

Given: a9 = 0

To prove: a29 = 2a19

Formula Used: nth term of an AP is:

an = a + (n - 1) d

Proof:

As a9 = 0

a + 8d = 0 [ eqn i]

Using the nth term formula i.e. an = a + (n - 1) d

Taking LHS

a29 = a + 28d

= (2 - 1) a + (36 - 8) d

= 2a - a + 36d - 8d

= 2a + 36d - (a + 8d )

= 2(a + 18d) - 0 [ using i]

= 2a19 [as a9 = a + 8d]

= RHS

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