Arithmetic Progression — Find the Term Position In the arithmetic sequence 5, 8, 11, 14, ... which term equals 320?

Introduction / Context:
Finding the position of a target value in an arithmetic progression (AP) is a standard problem. The AP formula connects term number, first term, and common difference.

Given Data / Assumptions:

• AP: 5, 8, 11, 14, ...
• First term a1 = 5
• Common difference d = 3
• Target value T = 320

Concept / Approach:
Use the nth-term formula for AP: a_n = a1 + (n − 1) * d. Solve for n when a_n = 320.

Step-by-Step Solution:
a_n = a1 + (n − 1) * d320 = 5 + (n − 1) * 3320 − 5 = (n − 1) * 3315 = (n − 1) * 3n − 1 = 315 / 3 = 105n = 105 + 1 = 106

Verification / Alternative check:
Compute the 106th term directly: a_106 = 5 + 105 * 3 = 5 + 315 = 320. Perfect match.

Why Other Options Are Wrong:

• 104th, 105th, 64th: Substituting these n values into the AP formula does not yield 320.

Common Pitfalls:
Arithmetic slips when isolating n. Always rearrange carefully and check with back-substitution.

Final Answer:
106th