Question

If 18^th and 11^th term of an A.P. are in the ratio 3 : 2, then its 21^st and 5^th terms are in the ratio

Options

•  3 : 2

•  3 : 1

• 1 : 3

• 2 : 3

Answer 1

In the given problem, we are given an A.P whose 18^th and 11^th term are in the ratio 3:2

We need to find the ratio of its 21^st and 5^th terms

Now, using the formula

a_n = a + ( n -1) d

Where,

a = first tem of the A.P

n = number of terms

d = common difference of the A.P

So,

a₁₈ = a + ( 18 - 1) d 

a₁₈ = a + 17d

Also,

a₁₁ = a + ( 11 -1 ) d

a₁₁ = a + 10 d

Thus,

               `(a_18)/(a_11) = 3/2`

     `( a + 17d) / ( a + 10 d) = 3/2`

  2 (a + 17 d ) = 3 ( a + 10d) 

      2a + 3ad = 3a + 30d

Further solving for a, we get

34 d - 30d = 3a - 2a

           4d = a                    .............(1)

Now,

a₂₁ = a + (21 - 1) d 

a₂₁ = a + 20 d

Also,

a₅ = a +( 5 -1) d

a₅ = a + 4d

So,

`a_21/a_5 = ( a + 20d ) /( a + 4d) `

Using (1) in the above equation, we get

`a_21/a_5 = ( 4d + 20 d) / (4d + 4d) `

`a_21/a_5 = (24d)/(8d)`

`a_21 / a_5 = 3/1`

Thus, the ratio of the 21^st and 5^th term is 3: 1 .