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The ratio of the 7th to the 3rd terms of an A.P. is 12 : 5. What is the ratio of the 13th to 4th term?

The ratio of the 7th to the 3rd terms of an A.P. is 12 : 5. What is the ratio of the 13th to 4th term? Correct Answer 10 : 3

Given,

 Ratio of the 7th to the 3rd terms of an = 12 : 5

Concept:/Formula:

Tn = a + (n – 1)d

a = first term

d = common difference

n = number of terms

Calculation:

T7 = a + (7 – 1) d

T7 = a + 6d

When n = 7

T3 = a + (3 – 1)d

T3 = a + 2d

According to the question

T7/T3 = 12/5

⇒ (a + 6d)/(a + 2d) = 12/5

⇒ 5a + 30d = 12a + 24d

⇒ 12a – 5a = 30d – 24d

⇒ 7a = 6d

⇒ a = 6d/7

T13 = a + (13 – 1) d

⇒ T13 = a + 12d

⇒ T13 = 6d/7 + 12d

⇒ T13 = 90d/7

T4 = a + (4 – 1) d

⇒ T4 = a + 3d

⇒ T4 = 6d/7 + 3d

⇒ T4 = 27d/7

T13 : T4 = 90d/7 : 27d/7

∴ T13 : T4 = 10 : 3

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