Let's solve each sum in parts. We will use the formula for the sum of the terms in a geometric progression:

$S_n = \frac{a(r^n - 1)}{r - 1}$

where:

• $a$ is the first term,
• $r$ is the common ratio,
• $n$ is the number of terms.

a) $\sum_{j=0}^{8} 3 \cdot 2^j$

This is a geometric progression with:

• First term $a = 3 \cdot 2^0 = 3$,
• Common ratio $r = 2$,
• Number of terms $n = 9$ (since it starts from $j = 0$ and ends at $j = 8$).

Using the sum formula:

$S_9 = \frac{3(2^9 - 1)}{2 - 1} = 3(2^9 - 1)$

Now calculate:

$2^9 = 512 \quad \Rightarrow \quad 2^9 - 1 = 511$

Thus:

$S_9 = 3 \times 511 = 1533$

So, the sum is 1533.

b) $\sum_{j=1}^{8} 2^j$

This is another geometric progression with:

• First term $a = 2^1 = 2$,
• Common ratio $r = 2$,
• Number of terms $n = 8$ (starting from $j = 1$ to $j = 8$).

Using the sum formula:

$S_8 = \frac{2(2^8 - 1)}{2 - 1} = 2(2^8 - 1)$

Now calculate:

$2^8 = 256 \quad \Rightarrow \quad 2^8 - 1 = 255$

Thus:

$S_8 = 2 \times 255 = 510$

So, the sum is 510.

c) $\sum_{j=2}^{8} (-3)^j$

This is a geometric progression with:

• First term $a = (-3)^2 = 9$,
• Common ratio $r = -3$,
• Number of terms $n = 7$ (since the series starts from $j = 2$ and ends at $j = 8$).

Using the sum formula:

$S_7 = \frac{9((-3)^7 - 1)}{-3 - 1} = \frac{9((-2187) - 1)}{-4}$

Now calculate:

$(-3)^7 = -2187 \quad \Rightarrow \quad (-3)^7 - 1 = -2188$

Thus:

$S_7 = \frac{9 \times (-2188)}{-4} = \frac{-19692}{-4} = 4923$

So, the sum is 4923.

d) $\sum_{j=0}^{8} 2 \cdot (-3)^j$

This is a geometric progression with:

• First term $a = 2 \cdot (-3)^0 = 2$,
• Common ratio $r = -3$,
• Number of terms $n = 9$ (from $j = 0$ to $j = 8$).

Using the sum formula:

$S_9 = \frac{2((-3)^9 - 1)}{-3 - 1} = \frac{2((-19683) - 1)}{-4}$

Now calculate:

$(-3)^9 = -19683 \quad \Rightarrow \quad (-3)^9 - 1 = -19684$

Thus:

$S_9 = \frac{2 \times (-19684)}{-4} = \frac{-39368}{-4} = 9842$

So, the sum is 9842.

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Summary of Results:

• a) 1533
• b) 510
• c) 4923
• d) 9842

Do you need further clarification on any of these steps or calculations?

Follow-up Questions:

1. How do the signs of terms in a geometric progression affect the sum?
2. What would happen to these sums if the common ratio were a fraction (e.g., $r = \frac{1}{2}$)?
3. How do you find the sum of an infinite geometric series?
4. Can you apply this formula to sums that involve alternating signs or negative terms?
5. How does changing the first term impact the sum of a geometric series?

Tip: When calculating powers of numbers, it’s helpful to keep track of exponents step-by-step to avoid mistakes.