Split 20 into four terms in AP with a product ratio 2:3: Divide 20 into four parts in arithmetic progression such that (1st × 4th) : (2nd × 3rd) = 2 : 3. Identify the four parts.

Introduction / Context:Again use the standard AP parameterization a−3d, a−d, a+d, a+3d with constraints on sum and a ratio of endpoint and middle products. The ratio eliminates the common factor (1/3)π type influences seen in geometric problems and focuses purely on algebraic relationships in a and d.

Given Data / Assumptions:

• Total sum = 20 ⇒ 4a = 20 ⇒ a = 5.
• AP terms as above.
• (T1*T4) : (T2*T3) = 2 : 3.

Concept / Approach:Use identities: T1*T4 = a^2 − 9d^2 and T2*T3 = a^2 − d^2. Form the ratio and solve for d.

Step-by-Step Solution:(a^2 − 9d^2) / (a^2 − d^2) = 2/3With a = 5: (25 − 9d^2) / (25 − d^2) = 2/33(25 − 9d^2) = 2(25 − d^2) ⇒ 75 − 27d^2 = 50 − 2d^225 = 25d^2 ⇒ d^2 = 1 ⇒ d = ±1Terms: 5−3 = 2, 5−1 = 4, 5+1 = 6, 5+3 = 8

Verification / Alternative check:Products: 2*8 = 16 and 4*6 = 24; the ratio is 16:24 = 2:3 as required.

Why Other Options Are Wrong:They either do not sum to 20 or are not an AP centered at 5 with common difference 1.

Common Pitfalls:Trying to brute-force options without noting the symmetric AP form around the average (sum/4).

Final Answer:2, 4, 6, 8