Question

Evaluate the given expression and show the steps, please.

Answer 1

80

Explanation:

To do this very simply, you need to input every value of k into the equation that the sigma notation asks of you and then sum them all together. You can do this question simply by inputting k = 1,2,3,4 into the equation and adding all the values together. (the bottom sigma value k=1 is the initial value of what you start with and the final value at the top is the value you need to get to by adding 1 to k each time and putting it in the equation every time i.e. think of it as 1 through 4 values you need to input into the equation and add together)

2(3^1-1) = 2

2(3^2-1) = 6

2(3^3-1) = 18

2(3^4-1) = 54

After adding them, you get 80.

Answer 2

s₄ = 80

Explanation:

`\displaystyle s_4=\sum^4_(k=1)2\left(3^(n-1)\right)`

The given sigma notation means the sum of the series with nth term 2(3ⁿ⁻¹), starting with n = 1 and ending with n = 4.

As 2(3ⁿ⁻¹) is exponential, the series is geometric.

`\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^(n-1)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}`

Comparing 2(3ⁿ⁻¹) with the general form of a geometric sequence:

• a = 2
• r = 3

The formula for the sum of the first n terms of a geometric series is:

`S_n=(a(1-r^n))/(1-r)`

Therefore, substitute a = 2, r = 3 and n = 4 into the sum formula to find the sum of the first 4 terms of the geometric series:

`\implies S_4=(2(1-3^4))/(1-3)`

`\implies S_4=(2(1-81))/(-2)`

`\implies S_4=(2(-80))/(-2)`

`\implies S_4=(-160)/(-2)`

`\implies S_4=80`

Therefore, the sum of the first 4 terms of the geometric series is 80.