Question

7.
How many arithmetic means should be
insterted between 5 and 83, so that the ratio
between the first and the last arithmetic mean
be 1: 7.​

Answer 1

Question :- How many arithmetic means should be insterted between 5 and 83, so that the ratio between the first and the last arithmetic mean be 1: 7. ?

Solution :-

given that,

• first term = a = 5
• last term = an = 83 .
• Let common difference = d .
• second term : second last term = 1 : 7 .

so,

→ second term = first term + common difference = (5 + d)

→ second last term = last term - common difference = (83 - d)

A/q,

→ (5 + d) / (83 - d) = 1/7

→ 7(5 + d) = 83 - d

→ 35 + 7d = 83 - d

→ 7d + d = 83 - 35

→ 8d = 48

→ d = 6 .

therefore,

→ Last term = an = 83

→ an = a + (n - 1)d

→ 5 + (n - 1)6 = 83

→ 5 + 6n - 6 = 83

→ 6n - 1 = 83

→ 6n = 83 + 1

→ 6n = 84

→ n = 14 .

hence,

→ Number of arithmetic means inserted between 5 and 84 = 14 - 2 = 12 (Ans.)