Find the $36th $ term of the series given below: $24,12,8,6,.$
- $\dfrac{2}{3}$
- $\dfrac{3}{2}$
- $\dfrac{3}{4}$
- $\dfrac{4}{3}$
The series is 24, 12, 8, 6. This can be written as 24/1, 24/2, 24/3, 24/4. The nth term is 24/n, so the 36th term is 24/36, which simplifies to 2/3.
The given series can be analyzed using successive differences or by converting it to the harmonic progression of its reciprocals. The reciprocals of the terms are 1 divided by 24, 1 divided by 12, 1 divided by 8, and 1 divided by 6, which simplifies to 1 divided by 24, 2 divided by 24, 3 divided by 24, and 4 divided by 24, forming an arithmetic progression with a common difference of 1 divided by 24. The 36th term of this reciprocal arithmetic progression is 36 divided by 24, which simplifies to 3 divided by 2. Inverting this gives the 36th term of the original series as 2 divided by 3.