The nth term of a sequence is given by 2$^{2n - 1}$. Find the sum of the first four terms.

1. Analyze the question

We are given the general formula for the $n^{th}$ term of a sequence: $T_n = 2^{2n - 1}$. We need to find the sum of the first four terms ($S_4$), which means calculating $T_1, T_2, T_3,$ and $T_4$ and adding them together.

2. Explain the concept

A sequence is an ordered list of numbers. To find the sum of the first $n$ terms, we evaluate the formula for each value of $n$ from 1 to $n$ and find their total.

• For $n=1$: $T_1 = 2^{2(1) - 1}$
• For $n=2$: $T_2 = 2^{2(2) - 1}$
• And so on.

3. Show the solution

Let's calculate each term individually:

• For $n = 1$: $T_1 = 2^{2(1) - 1} = 2^{2 - 1} = 2^1 = 2$

• For $n = 2$: $T_2 = 2^{2(2) - 1} = 2^{4 - 1} = 2^3 = 8$

• For $n = 3$: $T_3 = 2^{2(3) - 1} = 2^{6 - 1} = 2^5 = 32$

• For $n = 4$: $T_4 = 2^{2(4) - 1} = 2^{8 - 1} = 2^7 = 128$

Now, find the sum ($S_4$): $S_4 = T_1 + T_2 + T_3 + T_4$ $S_4 = 2 + 8 + 32 + 128$ $S_4 = 10 + 32 + 128$ $S_4 = 42 + 128$ $S_4 = 170$

4. Identify the answer

Comparing our result to the given options, the correct sum is 170, which corresponds to option D.

5. Exam Tip

When dealing with exponential sequences, always double-check your powers. A common mistake is to multiply the base by the exponent instead of raising the base to the power. Also, notice that this is a Geometric Progression (GP) where the first term $a = 2$ and the common ratio $r = 4$. You could also use the sum formula for a GP: $S_n = \frac{a(r^n - 1)}{r - 1}$. $S_4 = \frac{2(4^4 - 1)}{4 - 1} = \frac{2(256 - 1)}{3} = \frac{2(255)}{3} = 2 \times 85 = 170$.