# The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms. - Mathematics

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The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

#### Solution

Let a and d be the first term and common difference of an AP respectively.

Given that, a11 : a18 = 2 : 3

⇒ (a + 10d)/(a + 17d) = 2/3

⇒ 3a + 30d = 2a + 34d

⇒ a = 4d  .....(i)

Now, a_5 = a + 4d = 4d + 4d = 84d  ......[From equation (i)]

And a_21 = a + 20d = 4d + 20d = 24d  .....[From equation (i)]

∴ a_5 : a_21 = 8d : 24d = 1 : 3

Now, sum of the first five terms,

S_5 = 5/2[2a + (5 - 1)d]  .....[because S_n = n/2 [2a + (n - 1)d]]

= 5/2[2(4d) + 4d]  ....[From equation (i)]

= 5/4(8d + 4d)

= 5/2 xx 12d

= 30d

And sum of the first 21 terms,

S_21 = 21/2[2a + (21 - 1)d]

= 21/2[2(4d) + 20d]  ....[From equation (i)]

= 21/2(28d)

= 294d

So, ratio of the sum of the first five terms to the sum of the first 21 terms is

S5 : S21 = 30d : 29d = 5:49

Concept: Sum of First n Terms of an A.P.
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#### APPEARS IN

NCERT Mathematics Exemplar Class 10
Chapter 5 Arithematic Progressions
Exercise 5.4 | Q 6 | Page 57
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