# If the 9th Term of an A.P. is Zero, Then Prove that 29th Term is Double of 19th Term. - Algebra

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

#### Solution 1

t_n = a+ (n - 1)d

9th term i.e n = 9

:. t_9 = a+(9 -1)d

= a + 8d

It is given that t_9 = 0

∴ a + 8d = 0 ....(i)

29th term i.e t_29 where n = 29

:. t_29 = a + (29-1)d

t_29 = a + 28d ....(ii)

= (a + 8d) + 20d

= 0 + 20d      ...by equ (i)

:. t_29 = 20d ....(ii)

t_29 = a + (19 - 1)d

t_19 = a + 18d

= a + 8d + 10d

= 0 + 10d

t_19 = 10d .....(iiii)

by equation (ii) & (iii)

t_29 = 2t_19

#### Solution 2

In the given problem, the 9th term of an A.P. is zero.

Here, let us take the first term of the A.P as a and the common difference as d

So, as we know,

a_n = a + (n - 1)d

We get

a_9 = a + (9 - 1)d

0 = a + 8d

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

Now, we need to prove that 29th term is double of 19th term. So, let us first find the two terms.

For 19th term (n = 19)

a_19 = a + (19 - 1)d

= -8d  + 18d    (using 1)

= 10d

For 29th term (n = 29)

a_29 = a + (29 - 1)d

= - 8d + 28d

= 20d

= 2 x 10d

= 2 x a_19

(Using 1)

Therefore for the given A.P. the 29th term is double of the 19th term.

Hence proved.

Concept: General Term of an Arithmetic Progression
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Exercise 5.4 | Q 7 | Page 24