Question:
Consider the sequence: 72, 69, 66, ---. The numbers continue in the same pattern as long as they remain positive. What will be the maximum possible sum of the terms of this sequence?
Consider the sequence: 72, 69, 66, ---. The numbers continue in the same pattern as long as they remain positive. What will be the maximum possible sum of the terms of this sequence?
Show Hint
When an AP ends, the last term is usually the smallest positive value in the sequence.
Updated On: Jun 12, 2026
- 900
- 897
- 882
- 903
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
This is an arithmetic progression (AP) with first term \( a=72 \) and common difference \( d=-3 \). We need the sum of all positive terms.
Key Formula or Approach:
Last term \( a_n = a + (n-1)d \). We need \( a_n > 0 \).
Sum \( S_n = \frac{n}{2}(a + a_n) \).
Step 2: Detailed Explanation:
1. Find \( n \): \( 72 + (n-1)(-3) > 0 \implies 72 - 3n + 3 > 0 \implies 75 > 3n \implies n < 25 \).
2. Largest \( n = 24 \).
3. Find last term: \( a_{24} = 72 + (23)(-3) = 72 - 69 = 3 \).
4. Calculate Sum: \( S_{24} = \frac{24}{2}(72 + 3) = 12 \times 75 = 900 \).
Step 3: Final Answer:
The maximum sum is 900.
This is an arithmetic progression (AP) with first term \( a=72 \) and common difference \( d=-3 \). We need the sum of all positive terms.
Key Formula or Approach:
Last term \( a_n = a + (n-1)d \). We need \( a_n > 0 \).
Sum \( S_n = \frac{n}{2}(a + a_n) \).
Step 2: Detailed Explanation:
1. Find \( n \): \( 72 + (n-1)(-3) > 0 \implies 72 - 3n + 3 > 0 \implies 75 > 3n \implies n < 25 \).
2. Largest \( n = 24 \).
3. Find last term: \( a_{24} = 72 + (23)(-3) = 72 - 69 = 3 \).
4. Calculate Sum: \( S_{24} = \frac{24}{2}(72 + 3) = 12 \times 75 = 900 \).
Step 3: Final Answer:
The maximum sum is 900.
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