Question

The sum of the first 5 terms of a geometric progression is the same as the sum of the first 7 terms of the same progression. If the sum of the first 9 terms is 24, then the 4th term of the progression is

• A. 24
• B. -24
• C. -48
• D. 48

Hint:

The condition \( S_m = S_n \) (with \( m<n \)) for a GP implies that the sum of the terms from \( T_{m+1} \) to \( T_n \) is zero. This can often lead to a quick deduction about the common ratio, r.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem deals with the properties of a geometric progression (GP), specifically the sum of its terms.
Step 2: Key Formula or Approach:
Let the first term of the GP be 'a' and the common ratio be 'r'. The n-th term is \( T_n = ar^{n-1} \). The sum of the first n terms is \( S_n = \frac{a(r^n - 1)}{r-1} \) (for \( r \neq 1 \)). The given conditions are \( S_5 = S_7 \) and \( S_9 = 24 \). We need to find \( T_4 \).
Step 3: Detailed Explanation:
We are given the condition \( S_5 = S_7 \). This can be written as \( S_7 - S_5 = 0 \). The difference \( S_7 - S_5 \) is the sum of the terms from the 6th to the 7th, i.e., \( T_6 + T_7 \). So, \( T_6 + T_7 = 0 \). Using the formula for the n-th term: \[ ar^5 + ar^6 = 0 \] Factor out the common term \( ar^5 \): \[ ar^5(1 + r) = 0 \] For a non-trivial GP, we assume the first term \( a \neq 0 \). If \( r=0 \), the condition \( S_5=S_7=a \) holds, but then \( S_9=a=24 \) and \( T_4=ar^3=0 \), which is not an option. So, we must have: \[ 1 + r = 0 \implies r = -1 \] The common ratio of the GP is -1. This means the progression is an alternating series: \( a, -a, a, -a, \dots \) Now we use the second condition, \( S_9 = 24 \). Let's write out the sum of the first 9 terms with \( r=-1 \): \[ S_9 = a + a(-1) + a(-1)^2 + \dots + a(-1)^8 \] \[ S_9 = a - a + a - a + a - a + a - a + a \] The pairs of terms cancel out, and since there is an odd number of terms (9), the sum is simply the first term. \[ S_9 = a \] We are given \( S_9 = 24 \), so \( a = 24 \). Finally, we need to find the 4th term of the progression, \( T_4 \). \[ T_4 = ar^{4-1} = ar^3 \] Substitute the values of a and r we found: \[ T_4 = (24)(-1)^3 = (24)(-1) = -24 \] Step 4: Final Answer:
The 4th term of the progression is -24.