Question

What is the value of each of these sums of terms of a geometric progression?

$\begin{array}{r}\sum _{\mathrm{j}=0}^8 \left(3-2^{\mathrm{j}}\right)\\ \sum _{\mathrm{j}=1}^8 2^{\mathrm{j}}\\ \sum _{\mathrm{j}=2}^8 (-3{)}^{\mathrm{j}}\\ \sum _{\mathrm{j}=0}^8 2(-3{)}^{\mathrm{j}}\end{array}$

Short answer

Thus, the sum of terms of a geometric progression is given below:

1. 1533
2. 510
3. 4923
4. 9842

Step-by-step solution

Step 1:

(a) Given : $\sum _{j=0}^8 \left(3-2^j\right)$

Use the geometric series sum formula to compute the sum.

Note that ${\mathrm{a}}_0=3{\left(2\right)}^0\text{and}\mathrm{r}=2$

$\begin{array}{r}{a}_0\left[\frac{r^{n+1}-1}{r-1}\right]\\ =3\left[\frac{2^9-1}{2-1}\right]\\ =1533\end{array}$

Step 2:

Given: $:\sum _{j=1}^8 2^j$

Change the base, so that the geometric series sum formula can be used.

Let k = j - 1 so that j = k + 1

Now, ${\mathrm{a}}_0=2^1=2\mathrm{and}\mathrm{r}=2$

$\begin{array}{r}=\sum _{k=0}^7 2^{k+1}\\ =a_0\left[\frac{r^{n+1}-1}{r-1}\right]\\ =2\left[\frac{2^8-1}{2-1}\right]\\ =510\end{array}$

Step 3:

(c) Given $\sum _{j=2}^8 (-3{)}^j$

Change the base, so that the geometric series sum formula can be used.

Let k = j - 2 so that j = k + 2

Now, $a_0=(-3{)}^2=9\text{and}r=-3$

$\begin{array}{r}=a_0\left[\frac{r^{n+1}-1}{r-1}\right]\\ =9\left[\frac{(-3{)}^7-1}{-3-1}\right]\\ =4923\end{array}$

Step 4:

(d) Given $\sum _{j=0}^8 2\mathrm{\exists }(-3{)}^j$

Use the geometric series sum formula to compute the sum.

Note that $a_0=2(-3{)}^0=2\text{and}r=-3$

$\begin{array}{r}=a_0\left[\frac{r^{n+1}-1}{r-1}\right]\\ =2\left[\frac{(-3{)}^9-1}{-3-1}\right]\\ =9842\end{array}$