we have been asked to find the sum of the given geometric series
![\sum _{n=1}^4\left(\frac{1}{2}\right)^{n+1}](https://tex.z-dn.net/?f=%20%5Csum%20_%7Bn%3D1%7D%5E4%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bn%2B1%7D%20%20)
A geometric sequence has a constant ratio "r" and is given by
![r=\frac{a_{n+1}}{a_n}](https://tex.z-dn.net/?f=%20r%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%20)
![a_n=\left(\frac{1}{2}\right)^{n+1},\:a_{n+1}=\left(\frac{1}{2}\right)^{\left(n+1\right)+1}](https://tex.z-dn.net/?f=%20a_n%3D%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bn%2B1%7D%2C%5C%3Aa_%7Bn%2B1%7D%3D%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E%7B%5Cleft%28n%2B1%5Cright%29%2B1%7D%20)
![r=\frac{\left(\frac{1}{2}\right)^{\left(n+1\right)+1}}{\left(\frac{1}{2}\right)^{n+1}}=\frac{1}{2}](https://tex.z-dn.net/?f=%20r%3D%5Cfrac%7B%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E%7B%5Cleft%28n%2B1%5Cright%29%2B1%7D%7D%7B%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bn%2B1%7D%7D%3D%5Cfrac%7B1%7D%7B2%7D%20)
The first term of the sequence is
![a_1=\left(\frac{1}{2}\right)^{1+1}=\frac{1}{4}](https://tex.z-dn.net/?f=%20a_1%3D%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E%7B1%2B1%7D%3D%5Cfrac%7B1%7D%7B4%7D%20)
Sum of the sequence is given by the formula
![S_n=a_1\frac{1-r^n}{1-r}](https://tex.z-dn.net/?f=%20S_n%3Da_1%5Cfrac%7B1-r%5En%7D%7B1-r%7D%20)
Plug in the values we get
![S_4=\frac{1}{4}\cdot \frac{1-\left(\frac{1}{2}\right)^4}{1-\frac{1}{2}}](https://tex.z-dn.net/?f=%20S_4%3D%5Cfrac%7B1%7D%7B4%7D%5Ccdot%20%5Cfrac%7B1-%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E4%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%20)
On simplification we get
![S_4=\frac{15}{32}](https://tex.z-dn.net/?f=%20S_4%3D%5Cfrac%7B15%7D%7B32%7D%20)
Hence sum![=\frac{15}{32}](https://tex.z-dn.net/?f=%20%3D%5Cfrac%7B15%7D%7B32%7D%20)
Answer:
225 ft
Step-by-step explanation:
225 ft because, 30 inches divided by 2 is 15 and if 2 inches represents 15 feet, you have to do 15 x 15 which is 225! Hope this helps!
Answer:
I think it's the second answer, The head of state is always chosen by the citizens.
Step-by-step explanation:
I hope this helps!
Answer:
-1280
Step-by-step explanation:
Given a₁ = -10, n = 8, r =2
Required to find nᵗʰ term.
From the general formula: aₙ= a₁ (r)ⁿ⁻¹
Therefore nᵗʰ = -10(2)⁸⁻¹
= -10(2)⁷
= - 10(128)
= - 1280
Answer:
mark brainliest :)
Step-by-step explanation:
16 ^ 1.5 (or 3/2) = 64
64 ^ 0.5 (or 1/2) = 8
8 should be your answer