Question 33 of 44intermediate🔧 ApplyShort Answer4 marks

If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Correct Answer

The sum of first n terms is n².

Exercise: EXERCISE 5.3 | Q: 9 | (Page Page 21)
For More Understanding

Explanation

This question requires finding the sum formula Sn for an AP by first determining the first term 'a' and common difference 'd' using the given sum values. The formula Sn = n/2[2a + (n-1)d] from the textbook context is applied twice - once for S7 and once for S17 - to create two equations in 'a' and 'd'. Solving these simultaneous equations gives a = 1 and d = 2. Substituting these values back into the general sum formula yields the elegant result Sn = n².

Solution Steps

  1. Step 1: Apply the sum formula for S7. Given S7 = 49. Using Sn = n/2[2a + (n-1)d], we get: S7 = 7/2[2a + 6d] = 49. This simplifies to 2a + 6d = 14, or a + 3d = 7 ... (i)

  2. Step 2: Apply the sum formula for S17. Given S17 = 289. We get: S17 = 17/2[2a + 16d] = 289. This simplifies to 2a + 16d = 34, or a + 8d = 17 ... (ii)

  3. Step 3: Solve equations (i) and (ii). Subtracting (i) from (ii): (a + 8d) - (a + 3d) = 17 - 7. This gives 5d = 10, so d = 2. Substituting d = 2 in equation (i): a + 3(2) = 7, so a = 1.

  4. Step 4: Find the sum of first n terms. Using Sn = n/2[2a + (n-1)d] and substituting a = 1, d = 2: Sn = n/2[2(1) + (n-1)(2)] = n/2[2 + 2n - 2] = n/2×22 \times 2n = n².

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