Question

$x, 17, 3x - y^2 - 2, and 3x + y^2 - 30$ are four consecutive terms of an increasing arithmetic sequence. The sum of the four numbers is divisible by:

• A. 2
• B. 3
• C. 5
• D. 7
• E. 11

Hint:

When terms involve variables in an AP, always equate differences systematically. Once the variables are solved, check the sequence explicitly to confirm correctness.

Answer 1

The Correct Option is A

Step 1: Write the four terms of the arithmetic progression AP).
The given four consecutive terms are: \[ T_1 = x, T_2 = 17, T_3 = 3x - y^2 - 2, T_4 = 3x + y^2 - 30 \] Step 2: Use the property of an arithmetic progression.
In an AP, the difference between consecutive terms is the same. Thus, \[ T_2 - T_1 = T_3 - T_2 = T_4 - T_3 \] Step 3: Compute the first difference.
\[ T_2 - T_1 = 17 - x \] \[ T_3 - T_2 = (3x - y^2 - 2) - 17 = 3x - y^2 - 19 \] Equating: \[ 17 - x = 3x - y^2 - 19 ⇒ y^2 = 4x - 36 \] Step 4: Compute the next difference.
\[ T_4 - T_3 = (3x + y^2 - 30) - (3x - y^2 - 2) = 2y^2 - 28 \] \[ T_3 - T_2 = 3x - y^2 - 19 \] Equating: \[ 3x - y^2 - 19 = 2y^2 - 28 ⇒ 3x = 3y^2 - 9 ⇒ x = y^2 - 3 \] Step 5: Relating the two equations.
From Step 3: \( y^2 = 4x - 36 \) Substitute \( x = y^2 - 3 \): \[ y^2 = 4(y^2 - 3) - 36 ⇒ y^2 = 4y^2 - 12 - 36 \] \[ y^2 = 4y^2 - 48 ⇒ 3y^2 = 48 ⇒ y^2 = 16 \] So, \[ x = y^2 - 3 = 16 - 3 = 13 \] Step 6: Verify the sequence.
\[ T_1 = 13, T_2 = 17, T_3 = 3(13) - 16 - 2 = 21, T_4 = 3(13) + 16 - 30 = 25 \] So, the sequence is: \[ 13, \; 17, \; 21, \; 25 \] This is indeed an AP with common difference 4. Step 7: Find the sum.
\[ S = 13 + 17 + 21 + 25 = 76 \] Since \( 76 \div 2 = 38 \), the sum is divisible by 2. \[ \boxed{\text{The sum is divisible by 2 (Option (A)}} \]