Final answer:
To find the fifth term of the arithmetic progression (AP), determine the common difference by subtracting the second term from the first term. Solve the equation to find the value of 'a' and substitute it into the formula for the nth term to find the fifth term.
Step-by-step explanation:
To find the fifth term of the arithmetic progression (AP), we need to first determine the common difference. The common difference is the difference between any two consecutive terms in an AP. In this case, we can subtract the second term, 14, from the first term, 3²ᵃ⁻¹, to find the common difference.
3²ᵃ⁻¹ - 14 = 3⁴ᵃ⁻¹ - 14
The given sequence, with terms 3²ᵃ⁻¹, 14, 3⁴ᵃ⁻¹ representing the first three terms of an arithmetic progression (AP), requires finding the common difference (d). By setting up equations with the second and third terms, involving powers of 3 and d, you can solve for d.
Once d is determined, the fifth term (5a) is found by adding 4 times the common difference to the first term. Comparing 5a to the provided choices (A, B, C, D) identifies the correct fifth term in the arithmetic progression, offering a succinct insight into solving this mathematical problem.
We can then solve this equation to find the value of 'a'. Once we know 'a', we can substitute it into the formula for the nth term of an AP to find the fifth term.