If the arithmetic mean and geometric mean of the pᵗʰ and qᵗʰ terms of the
sequence -16, 8, -4, 2, ...... satisfy the equation
4x² - 9x + 5 = 0, then p + q is equal to __________.

Solution

Maths sir · 4 steps

Decide which root is the AM and which is the GM, then use only the product of the two terms.

1. Roots first. 4x² - 9x + 5 = (x - 1)(4x - 5), so the roots are 1 and 5/4. Since AM ≥ GM, the AM is 5/4 and the GM is 1.
2. The sequence. Here a = -16 and r = -1/2, so Tₙ = -16(-1/2)ⁿ⁻¹. The GM is 1, so Tₚ T_q = GM² = 1.

   TₚT_q = 256(-1/2)^(p+q-2) = 1 ⇒ (-1/2)^(p+q-2) = 1/256 = (1/2)⁸

3. Check it works. So p + q - 2 = 8. The terms with sum 2 × 5/4 = 5/2 and product 1 are 2 and 1/2, which are T₄ and T₆. The pair fits.
4. So. p + q = 8 + 2 = 10.

The common slip: 8. Writing each term as arⁿ instead of arⁿ⁻¹ drops the -2 from the exponent and gives 8, not 10.

The idea to keep. When the AM and GM are the roots of a quadratic, the larger root is the AM. The product of two GP terms depends only on p + q, so square the GM and compare powers.

Answer from the NTA final answer key. Solution by PrepWiser.