Sequences & Series
Sum of Arithmetic Progression
Grade 11
Question:
<p>A total prize of ₹8000 is to be distributed among 16 teams. If the first place team gets ₹275 and there is a common difference of prize amounts between consecutive places, find the prize for the team that comes first.</p>
Step-by-Step Solution
Key Concept: Use the arithmetic progression sum formula to find the common difference, then calculate the specific term requested.
<p><strong>Solution:</strong></p><p>Given: $n = 16$, $S_n = 8000$, and first term $a = 275$ (considering last place team as first term).</p><p>Using the sum formula for AP:</p><p>$$S_n = \frac{n}{2}[2a + (n-1)d]$$</p><p>$$8000 = \frac{16}{2}[2(275) + (16-1)d]$$</p><p>$$8000 = 8[550 + 15d]$$</p><p>$$1000 = 550 + 15d$$</p><p>$$15d = 450$$</p><p>$$d = 30$$</p><p>The 16th term (first place team) is:</p><p>$$T_{16} = 275 + (16-1) \times 30 = 275 + 450 = 725$$</p>
Correct Answer: 725