Geometric sequence is characterized by a common ratio
(1) Sum of first 5 terms
The first term of the sequence is:
[tex]\mathbf{a = 3}[/tex]
The common ratio (r) is:
[tex]\mathbf{r = 6 \div 3 = 2}[/tex]
The sum of n terms is calculated using:
[tex]\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}[/tex]
So, we have:
[tex]\mathbf{S_5 = \frac{3 \times (2^5 - 1)}{2 - 1}}[/tex]
[tex]\mathbf{S_5 = \frac{93}{1}}[/tex]
[tex]\mathbf{S_5 = 93}[/tex]
Hence, the sum of the first five terms is 93
(2) Sum of first 5 terms
The first term of the sequence is:
[tex]\mathbf{a = 14}[/tex]
The common ratio (r) is:
[tex]\mathbf{r = 3}[/tex]
The sum of n terms is calculated using:
[tex]\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}[/tex]
So, we have:
[tex]\mathbf{S_5 = \frac{14 \times (3^5 - 1)}{3 - 1}}[/tex]
[tex]\mathbf{S_5 = \frac{3388}{2}}[/tex]
[tex]\mathbf{S_5 = 1694}[/tex]
Hence, the sum of the first five terms is 1694
(3) Sum of first n terms
The first term of the sequence is:
[tex]\mathbf{a = 318}[/tex]
The common ratio (r) is:
[tex]\mathbf{r = \frac12}[/tex]
The sum of n terms is calculated using:
[tex]\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}[/tex]
So, we have:
[tex]\mathbf{S_n = \frac{318 \times (1 - \frac 12^n)}{1 - \frac 12}}[/tex]
[tex]\mathbf{S_n = \frac{318 \times (1 - \frac 12^n)}{\frac 12}}[/tex]
[tex]\mathbf{S_n = 636 (1 - \frac 12^n)}[/tex]
Hence, the sum of the first n terms is [tex]\mathbf{ 636 (1 - \frac 12^n)}[/tex]
(4) The first term
The sum of the first 7th term of the sequence is:
[tex]\mathbf{S_7 = 547}[/tex]
The common ratio (r) is:
[tex]\mathbf{r = -3}[/tex]
The sum of n terms is calculated using:
[tex]\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}[/tex]
So, we have:
[tex]\mathbf{547 = \frac{a(1 - (-3)^7)}{1 - -3}}[/tex]
[tex]\mathbf{547 = \frac{a(2188)}{4}}[/tex]
Multiply both sides by 4
[tex]\mathbf{2188= a(2188)}[/tex]
Divide both sides by 2188
[tex]\mathbf{1= a}[/tex]
Rewrite as:
[tex]\mathbf{a = 1}[/tex]
Hence, the first term is 1
(5) Find the 7th term
The first term of the sequence is:
[tex]\mathbf{a=2}[/tex]
The common ratio (r) is:
[tex]\mathbf{r=3}[/tex]
The nth term of a geometric sequence is:
[tex]\mathbf{T_n = ar^{n -1}}[/tex]
So, we have:
[tex]\mathbf{T_7 = 2 \times 3^{7 -1}}[/tex]
[tex]\mathbf{T_7 = 1458}[/tex]
Hence, the seventh term is 1458
(6) Sum of geometric sequence
The first term of the sequence is:
[tex]\mathbf{a=2}[/tex]
The common ratio of the sequence is:
[tex]\mathbf{r = 6 \div 2 = 3}[/tex]
The number of terms is:
[tex]\mathbf{n = 5}[/tex]
The sum of n terms is calculated using:
[tex]\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}[/tex]
So, we have:
[tex]\mathbf{S_5 = \frac{2 \times (3^5 - 1)}{3 - 1}}[/tex]
[tex]\mathbf{S_5 = \frac{484}{2}}[/tex]
[tex]\mathbf{S_5 = 242}[/tex]
Hence, the sum of the first five terms is 242
(7) The first term
The sum of the first five terms is given as:
[tex]\mathbf{S_5 = 341}[/tex]
The common ratio is:
[tex]\mathbf{r = 4}[/tex]
The sum of n terms is calculated using:
[tex]\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}[/tex]
So, we have:
[tex]\mathbf{341 = \frac{a \times (4^5 - 1)}{4 - 1}}[/tex]
[tex]\mathbf{341 = \frac{a \times 1023}{3}}[/tex]
Solve for a
[tex]\mathbf{a = \frac{3 \times 341}{1023}}[/tex]
[tex]\mathbf{a = 1}[/tex]
Hence, the first terms is 1
(8) Sum to infinite
The first term of the sequence is:
[tex]\mathbf{a = 192}[/tex]
The common ratio (r) is:
[tex]\mathbf{r = \frac 14}[/tex]
The sum to infinite is:
[tex]\mathbf{S_{\infty} = \frac{a}{1 - r}}[/tex]
So, we have:
[tex]\mathbf{S_{\infty} = \frac{192}{1 - 1/4}}[/tex]
[tex]\mathbf{S_{\infty} = \frac{192}{3/4}}[/tex]
[tex]\mathbf{S_{\infty} = 256}[/tex]
Hence, the sum to infinite is 256
Read more about geometric sequence at:
https://brainly.com/question/18109692
