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Question
If 18th and 11th term of an A.P. are in the ratio 3 : 2, then its 21st and 5th terms are in the ratio
Options
3 : 2
3 : 1
1 : 3
2 : 3
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Solution
In the given problem, we are given an A.P whose 18th and 11th term are in the ratio 3:2
We need to find the ratio of its 21st and 5th terms
Now, using the formula
an = 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 21st and 5th term is 3: 1 .
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