Answer:
The five arithmetic means between 14 and 86 are 26,38,50,62,74
Step-by-step explanation:
We need to find five arithmetic means between 14 and 86
The formula used to find arithmetic mean is: ![a_{n}=a_{1} +(n-1)d](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d)
Where n is the number of terms and d is the common difference
We are given 1st and 7th term.
Using 7th term to find value of d
![a_{n}=a_{1} +(n-1)d\\a_{7}=14 +(7-1)d\\86=14+6d\\86-14=6d\\72=6d\\d=\frac{72}{6}\\d=12](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d%5C%5Ca_%7B7%7D%3D14%20%2B%287-1%29d%5C%5C86%3D14%2B6d%5C%5C86-14%3D6d%5C%5C72%3D6d%5C%5Cd%3D%5Cfrac%7B72%7D%7B6%7D%5C%5Cd%3D12)
So, difference between each term d is 12
Now finding five arithmetic terms i.e a₂, a₃, a₄, a₅, a₆
Using the formula
to find terms
Finding term a₂
![a_{n}=a_{1} +(n-1)d\\a_{2}=14 +(2-1)12\\a_{2}=14 +12\\a_{2}=26](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d%5C%5Ca_%7B2%7D%3D14%20%2B%282-1%2912%5C%5Ca_%7B2%7D%3D14%20%2B12%5C%5Ca_%7B2%7D%3D26)
Finding term a₃
![a_{n}=a_{1} +(n-1)d\\a_{3}=14 +(3-1)12\\a_{3}=14 +2*12\\a_{3}=38](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d%5C%5Ca_%7B3%7D%3D14%20%2B%283-1%2912%5C%5Ca_%7B3%7D%3D14%20%2B2%2A12%5C%5Ca_%7B3%7D%3D38)
Finding term a₄
![a_{n}=a_{1} +(n-1)d\\a_{4}=14 +(4-1)12\\a_{4}=14 +3*12\\a_{4}=50](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d%5C%5Ca_%7B4%7D%3D14%20%2B%284-1%2912%5C%5Ca_%7B4%7D%3D14%20%2B3%2A12%5C%5Ca_%7B4%7D%3D50)
Finding term a₅
![a_{n}=a_{1} +(n-1)d\\a_{5}=14 +(5-1)12\\a_{5}=14 +4*12\\a_{5}=62](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d%5C%5Ca_%7B5%7D%3D14%20%2B%285-1%2912%5C%5Ca_%7B5%7D%3D14%20%2B4%2A12%5C%5Ca_%7B5%7D%3D62)
Finding term a₆
![a_{n}=a_{1} +(n-1)d\\a_{6}=14 +(6-1)12\\a_{6}=14 +5*12\\a_{6}=74](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%20%2B%28n-1%29d%5C%5Ca_%7B6%7D%3D14%20%2B%286-1%2912%5C%5Ca_%7B6%7D%3D14%20%2B5%2A12%5C%5Ca_%7B6%7D%3D74)
So, the five arithmetic means between 14 and 86 are 26,38,50,62,74