Question

For a sequence if \( S_n = \dfrac{5^n - 2^n}{2^n} \), then its fourth term is

• A. \( \dfrac{375}{16} \)
• B. \( \dfrac{375}{8} \)
• C. \( \dfrac{251}{8} \)
• D. \( \dfrac{251}{16} \)

Hint:

When the sum of a sequence is given, individual terms can be found using \( a_n = S_n - S_{n-1} \).

Answer 1

The Correct Option is A

Step 1: Write the given formula.
The sum of the first \( n \) terms is given as \[ S_n = \dfrac{5^n - 2^n}{2^n} \]

Step 2: Find the fourth term using \( a_n = S_n - S_{n-1} \).
\[ S_4 = \dfrac{5^4 - 2^4}{2^4} = \dfrac{625 - 16}{16} = \dfrac{609}{16} \] \[ S_3 = \dfrac{5^3 - 2^3}{2^3} = \dfrac{125 - 8}{8} = \dfrac{117}{8} \]

Step 3: Compute the fourth term.
\[ a_4 = S_4 - S_3 = \dfrac{609}{16} - \dfrac{234}{16} = \dfrac{375}{16} \]

Step 4: Conclusion.
The fourth term of the sequence is \[ \boxed{\dfrac{375}{16}} \]