Geometric sequence is characterized by a common ratio
<u>(1) Sum of first 5 terms</u>
The first term of the sequence is:
![\mathbf{a = 3}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%203%7D)
The common ratio (r) is:
![\mathbf{r = 6 \div 3 = 2}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%206%20%5Cdiv%203%20%3D%202%7D)
The sum of n terms is calculated using:
![\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Br%20-%201%7D%7D)
So, we have:
![\mathbf{S_5 = \frac{3 \times (2^5 - 1)}{2 - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20%5Cfrac%7B3%20%5Ctimes%20%282%5E5%20-%201%29%7D%7B2%20-%201%7D%7D)
![\mathbf{S_5 = \frac{93}{1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20%5Cfrac%7B93%7D%7B1%7D%7D)
![\mathbf{S_5 = 93}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%2093%7D)
Hence, the sum of the first five terms is 93
<u>(2) Sum of first 5 terms</u>
The first term of the sequence is:
![\mathbf{a = 14}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%2014%7D)
The common ratio (r) is:
![\mathbf{r = 3}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%203%7D)
The sum of n terms is calculated using:
![\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Br%20-%201%7D%7D)
So, we have:
![\mathbf{S_5 = \frac{14 \times (3^5 - 1)}{3 - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20%5Cfrac%7B14%20%5Ctimes%20%283%5E5%20-%201%29%7D%7B3%20-%201%7D%7D)
![\mathbf{S_5 = \frac{3388}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20%5Cfrac%7B3388%7D%7B2%7D%7D)
![\mathbf{S_5 = 1694}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%201694%7D)
Hence, the sum of the first five terms is 1694
<u>(3) Sum of first n terms</u>
The first term of the sequence is:
![\mathbf{a = 318}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%20318%7D)
The common ratio (r) is:
![\mathbf{r = \frac12}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%20%5Cfrac12%7D)
The sum of n terms is calculated using:
![\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7Ba%281%20-%20r%5En%29%7D%7B1%20-%20r%7D%7D)
So, we have:
![\mathbf{S_n = \frac{318 \times (1 - \frac 12^n)}{1 - \frac 12}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7B318%20%5Ctimes%20%281%20-%20%5Cfrac%2012%5En%29%7D%7B1%20-%20%5Cfrac%2012%7D%7D)
![\mathbf{S_n = \frac{318 \times (1 - \frac 12^n)}{\frac 12}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7B318%20%5Ctimes%20%281%20-%20%5Cfrac%2012%5En%29%7D%7B%5Cfrac%2012%7D%7D)
![\mathbf{S_n = 636 (1 - \frac 12^n)}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20636%20%281%20-%20%5Cfrac%2012%5En%29%7D)
Hence, the sum of the first n terms is ![\mathbf{ 636 (1 - \frac 12^n)}](https://tex.z-dn.net/?f=%5Cmathbf%7B%20636%20%281%20-%20%5Cfrac%2012%5En%29%7D)
<u>(4) The first term</u>
The sum of the first 7th term of the sequence is:
![\mathbf{S_7 = 547}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_7%20%3D%20547%7D)
The common ratio (r) is:
![\mathbf{r = -3}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%20-3%7D)
The sum of n terms is calculated using:
![\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7Ba%281%20-%20r%5En%29%7D%7B1%20-%20r%7D%7D)
So, we have:
![\mathbf{547 = \frac{a(1 - (-3)^7)}{1 - -3}}](https://tex.z-dn.net/?f=%5Cmathbf%7B547%20%3D%20%5Cfrac%7Ba%281%20-%20%28-3%29%5E7%29%7D%7B1%20-%20-3%7D%7D)
![\mathbf{547 = \frac{a(2188)}{4}}](https://tex.z-dn.net/?f=%5Cmathbf%7B547%20%3D%20%5Cfrac%7Ba%282188%29%7D%7B4%7D%7D)
Multiply both sides by 4
![\mathbf{2188= a(2188)}](https://tex.z-dn.net/?f=%5Cmathbf%7B2188%3D%20a%282188%29%7D)
Divide both sides by 2188
![\mathbf{1= a}](https://tex.z-dn.net/?f=%5Cmathbf%7B1%3D%20a%7D)
Rewrite as:
![\mathbf{a = 1}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%201%7D)
Hence, the first term is 1
<u>(5) Find the 7th term</u>
The first term of the sequence is:
![\mathbf{a=2}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%3D2%7D)
The common ratio (r) is:
![\mathbf{r=3}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%3D3%7D)
The nth term of a geometric sequence is:
![\mathbf{T_n = ar^{n -1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BT_n%20%3D%20ar%5E%7Bn%20-1%7D%7D)
So, we have:
![\mathbf{T_7 = 2 \times 3^{7 -1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BT_7%20%3D%202%20%5Ctimes%203%5E%7B7%20-1%7D%7D)
![\mathbf{T_7 = 1458}](https://tex.z-dn.net/?f=%5Cmathbf%7BT_7%20%3D%201458%7D)
Hence, the seventh term is 1458
<u />
<u>(6) Sum of geometric sequence</u>
The first term of the sequence is:
![\mathbf{a=2}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%3D2%7D)
The common ratio of the sequence is:
![\mathbf{r = 6 \div 2 = 3}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%206%20%5Cdiv%202%20%3D%203%7D)
The number of terms is:
![\mathbf{n = 5}](https://tex.z-dn.net/?f=%5Cmathbf%7Bn%20%3D%205%7D)
The sum of n terms is calculated using:
![\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Br%20-%201%7D%7D)
So, we have:
![\mathbf{S_5 = \frac{2 \times (3^5 - 1)}{3 - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20%5Cfrac%7B2%20%5Ctimes%20%283%5E5%20-%201%29%7D%7B3%20-%201%7D%7D)
![\mathbf{S_5 = \frac{484}{2}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20%5Cfrac%7B484%7D%7B2%7D%7D)
![\mathbf{S_5 = 242}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20242%7D)
Hence, the sum of the first five terms is 242
<u>(7) The first term</u>
The sum of the first five terms is given as:
![\mathbf{S_5 = 341}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_5%20%3D%20341%7D)
The common ratio is:
![\mathbf{r = 4}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%204%7D)
The sum of n terms is calculated using:
![\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Br%20-%201%7D%7D)
So, we have:
![\mathbf{341 = \frac{a \times (4^5 - 1)}{4 - 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7B341%20%3D%20%5Cfrac%7Ba%20%5Ctimes%20%284%5E5%20-%201%29%7D%7B4%20-%201%7D%7D)
![\mathbf{341 = \frac{a \times 1023}{3}}](https://tex.z-dn.net/?f=%5Cmathbf%7B341%20%3D%20%5Cfrac%7Ba%20%5Ctimes%201023%7D%7B3%7D%7D)
Solve for a
![\mathbf{a = \frac{3 \times 341}{1023}}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%20%5Cfrac%7B3%20%5Ctimes%20341%7D%7B1023%7D%7D)
![\mathbf{a = 1}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%201%7D)
Hence, the first terms is 1
<u>(8) Sum to infinite</u>
The first term of the sequence is:
![\mathbf{a = 192}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba%20%3D%20192%7D)
The common ratio (r) is:
![\mathbf{r = \frac 14}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%20%3D%20%5Cfrac%2014%7D)
The sum to infinite is:
![\mathbf{S_{\infty} = \frac{a}{1 - r}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_%7B%5Cinfty%7D%20%3D%20%5Cfrac%7Ba%7D%7B1%20-%20r%7D%7D)
So, we have:
![\mathbf{S_{\infty} = \frac{192}{1 - 1/4}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_%7B%5Cinfty%7D%20%3D%20%5Cfrac%7B192%7D%7B1%20-%201%2F4%7D%7D)
![\mathbf{S_{\infty} = \frac{192}{3/4}}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_%7B%5Cinfty%7D%20%3D%20%5Cfrac%7B192%7D%7B3%2F4%7D%7D)
![\mathbf{S_{\infty} = 256}](https://tex.z-dn.net/?f=%5Cmathbf%7BS_%7B%5Cinfty%7D%20%3D%20256%7D)
Hence, the sum to infinite is 256
Read more about geometric sequence at:
brainly.com/question/18109692