Question 44 of 51advanced🔧 ApplyNumerical4 marks

The sum of the 4th4^{th} and 8th8^{th} terms of an AP is 24 and the sum of the 6th6^{th} and 10th10^{th} terms is 44. Find the first three terms of the AP.

Correct Answer

-13, -8, -3

Exercise: END-OF-CHAPTER EXERCISES | Q: 8 | (Page 22)
For More Understanding

Explanation

The question requires finding the first three terms of an AP using the given sums of specific terms. Using the formula for the nth term of an AP, t_n = a + (n-1)d, we translate the given conditions into a pair of linear equations in 'a' and 'd'. Solving these equations yields the first term (a) and the common difference (d), which are then used to calculate the first three terms of the AP.

Solution Steps

  1. Step 1: Using the formula for the nth term of an AP, t_n = a + (n-1)d, write the expressions for the given terms. The 4th term is a + 3d and the 8th term is a + 7d.

  2. Step 2: Form the first equation from the given sum: (a + 3d) + (a + 7d) = 24. This simplifies to 2a + 10d = 24, or a + 5d = 12 (Equation 1).

  3. Step 3: Form the second equation from the second given sum. The 6th term is a + 5d and the 10th term is a + 9d. So, (a + 5d) + (a + 9d) = 44. This simplifies to 2a + 14d = 44, or a + 7d = 22 (Equation 2).

  4. Step 4: Solve the system of equations. Subtract Equation 1 from Equation 2: (a + 7d) - (a + 5d) = 22 - 12. This gives 2d = 10, so d = 5.

  5. Step 5: Substitute d = 5 into Equation 1: a + 5(5) = 12, which gives a + 25 = 12, so a = -13.

  6. Step 6: Find the first three terms of the AP. The first term is a = -13. The second term is a + d = -13 + 5 = -8. The third term is a + 2d = -13 + 10 = -3. The first three terms are -13, -8, -3.

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