If log ₑ a, log ₑ b, log ₑ c are in an A.P. and log ₑ a-log ₑ 2 b, log ₑ 2 b-log ₑ 3 c, log ₑ 3 c -log ₑ a are also in an A.P, then a: b: c is equal to

Solution

Maths sir · 4 steps

Logs in A.P. mean the numbers are in G.P., and the second A.P. sums to zero, so its middle term is zero.

1. First A.P. ln a, ln b, ln c in A.P. gives 2 ln b = ln a + ln c, so b² = ac. That means a, b, c are in G.P.
2. Add the second A.P. The three terms add to ln a/2b + ln 2b/3c + ln 3c/a = ln 1 = 0. In any A.P. the sum of three terms is 3 times the middle term, so the middle term is 0.

   ln 2b - ln 3c = 0 ⇒ 2b = 3c ⇒ b : c = 3 : 2

3. Find a. Take b = 6, c = 4. Then b² = ac gives a = 36/4 = 9.
4. So. a : b : c = 9 : 6 : 4, which is option (b). Check: 6² = 9 × 4 and 2 × 6 = 3 × 4.

The idea to keep. Logs in A.P. means the numbers are in G.P. And when three terms in A.P. add to zero, the middle term must be zero. Spotting that sum saves the algebra.

Answer from the NTA final answer key. Solution by PrepWiser.