Math Snap

STEP 1

What is this asking?
We need to rewrite a sum using the fancy sigma notation, and then calculate what it equals!
Watch out!
Make sure to start and end at the correct values of $k$!

STEP 2

1. Rewrite the sum using sigma notation.
2. Calculate the sum.

STEP 3

The general term in our sum is $2^{k-1} + 4$.
This tells us what each piece of the sum looks like.

STEP 4

We're told that $k$ goes from $\mathbf{2}$ to $\mathbf{5}$, inclusive.
These are our starting and ending values.

STEP 5

Using the sigma notation, we can write the sum as:
$$\sum_{k=2}^{5} (2^{k-1} + 4)$$ So we're adding up the expression $2^{k-1} + 4$ for each value of $k$ from $2$ to $5$.

STEP 6

Let's write out each term of the sum:
$$\sum_{k=2}^{5} (2^{k-1} + 4) = (2^{2-1} + 4) + (2^{3-1} + 4) + (2^{4-1} + 4) + (2^{5-1} + 4)$$

STEP 7

Let's simplify those exponents:
$$(2^{1} + 4) + (2^{2} + 4) + (2^{3} + 4) + (2^{4} + 4)$$

STEP 8

Calculating the powers of $2$, we get:
$$(2 + 4) + (4 + 4) + (8 + 4) + (16 + 4)$$

STEP 9

Adding the numbers inside the parentheses gives us:
$$6 + 8 + 12 + 20$$

STEP 10

Finally, adding all the terms together, we get:
$$6 + 8 + 12 + 20 = \mathbf{46}$$

The sum in sigma notation is $\sum_{k=2}^{5} (2^{k-1} + 4)$, and its value is $\mathbf{46}$.