In a G.P., the \(3^{\text {rd}}\) term is 24 and the \(6^{\text {th}}\) term is 192. What is its \(10^{\text {th}}\) term?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3072
In a G.P. the \(n^{\text{th}}\) term is \(ar^{n-1}\). Dividing the 6th term by the 3rd cancels \(a\) and leaves \(r^3\).
3rd term: \(ar^2=24\) (1)
6th term: \(ar^5=192\) (2)
Dividing (2) by (1), \(a\) cancels: \(r^3=8\), so \(r=2\)
From (1): \(4a=24\), so \(a=6\)
10th term \(=ar^9=6\times2^9=6\times512=3072\)
Shortcut: The 10th term is 4 steps after the 6th, so it is \(192\times r^4=192\times16=3072\).