Answer:
[tex]3 + 12 + 48 + 192 + 768 = \sum\limits^4_{n=0} 3 * 4^n[/tex]
[tex]4 + 32 + 256 + 2048 + 16384 = \sum\limits^4_{n=0} 4 * 8^n[/tex]
[tex]2 + 6 + 18 + 54 + 162 = \sum\limits^4_{n=0} 2* 3^n[/tex]
[tex]3 + 15 + 75 + 375 + 1875 = \sum\limits^4_{n=0} 3* 5^n[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required
Match equivalent expressions
Solving (a):
[tex]3 + 12 + 48 + 192 + 768[/tex]
The expression can be written as:
[tex]3 \to 3*4^{0[/tex] --- 0
[tex]12 \to 3 * 4^{1[/tex] ---- 1
[tex]48 \to 3 * 4^{2[/tex] --- 2
[tex]192 \to 3 * 4^{3[/tex] ---- 3
[tex]768 \to 3 * 4^{4[/tex] ---- 4
For the nth term, the expression is:
[tex]Term = 3 * 4^{n[/tex] ---- n
So, the summation is:
[tex]3 + 12 + 48 + 192 + 768 = \sum\limits^4_{n=0} 3 * 4^n[/tex]
Solving (b):
[tex]4 + 32 + 256 + 2048 + 16384[/tex]
The expression can be written as:
[tex]4 \to 4 * 8^0[/tex] --- 0
[tex]32 \to 4 * 8^1[/tex] ---- 1
[tex]256 \to 4 * 8^2[/tex] --- 2
[tex]2048 \to 4 * 8^3[/tex] ---- 3
[tex]16384 \to 4 * 8^4[/tex] ---- 4
For the nth term, the expression is:
[tex]Term \to 4 * 8^n[/tex] ---- n
So, the summation is:
[tex]4 + 32 + 256 + 2048 + 16384 = \sum\limits^4_{n=0} 4 * 8^n[/tex]
Solving (c):
[tex]2 + 6 + 18 + 54 + 162[/tex]
The expression can be written as:
[tex]2 \to 2 * 3^0[/tex] --- 0
[tex]6 \to 2 * 3^1[/tex] ---- 1
[tex]18 \to 2 * 3^2[/tex] --- 2
[tex]54 \to 2 * 3^3[/tex] ---- 3
[tex]162 \to 2 * 3^4[/tex] ---- 4
For the nth term, the expression is:
[tex]Term \to 2 * 3^n[/tex] ---- n
So, the summation is:
[tex]2 + 6 + 18 + 54 + 162 = \sum\limits^4_{n=0} 2* 3^n[/tex]
Solving (d):
[tex]3 + 15 + 75 + 375 + 1875[/tex]
The expression can be written as:
[tex]3 \to 3 * 5^0[/tex] --- 0
[tex]15 \to 3 * 5^1[/tex] ---- 1
[tex]75 \to 3 * 5^2[/tex] --- 2
[tex]375 \to 3 * 5^3[/tex] ---- 3
[tex]1875 \to 3 * 5^4[/tex] ---- 4
For the nth term, the expression is:
[tex]Term \to 3 * 5^n[/tex] ---- n
So, the summation is:
[tex]3 + 15 + 75 + 375 + 1875 = \sum\limits^4_{n=0} 3* 5^n[/tex]
