Answer:
![a = -9](https://tex.z-dn.net/?f=a%20%3D%20-9)
![b = 4](https://tex.z-dn.net/?f=b%20%3D%204)
Step-by-step explanation:
Given
Sequence: a+3b, a+7b, a+11b
2nd term = 19
5th term = 67
Required
Find a and b
First, the 5th term needs to be calculated;
Using formula for Arithmetic Progression (AP), the formula goes thus
![T_n = T_1 + (n - 1)d](https://tex.z-dn.net/?f=T_n%20%3D%20T_1%20%2B%20%28n%20-%201%29d)
Where n = 5
T_1 = a + 3b ------------ FIrst term
--- Difference between two successive terms
![d = a + 7b - (a + 3b)](https://tex.z-dn.net/?f=d%20%3D%20a%20%2B%207b%20-%20%28a%20%2B%203b%29)
![d = a + 7b -a - 3b](https://tex.z-dn.net/?f=d%20%3D%20a%20%2B%207b%20-a%20-%203b)
![d = a - a + 7b - 3b](https://tex.z-dn.net/?f=d%20%3D%20a%20-%20a%20%2B%207b%20-%203b)
![d = 4b](https://tex.z-dn.net/?f=d%20%3D%204b)
So,
becomes
![T_5 = a + 3b + (5 - 1)4b](https://tex.z-dn.net/?f=T_5%20%3D%20a%20%2B%203b%20%2B%20%285%20-%201%294b)
![T_5 = a + 3b + (4)4b](https://tex.z-dn.net/?f=T_5%20%3D%20a%20%2B%203b%20%2B%20%284%294b)
![T_5 = a + 3b + 16b](https://tex.z-dn.net/?f=T_5%20%3D%20a%20%2B%203b%20%2B%2016b)
![T_5 = a + 19b](https://tex.z-dn.net/?f=T_5%20%3D%20a%20%2B%2019b)
Now that we have values for 2nd and 5th term;
From the second, T2 = 19 and T5 = 67
This gives
--- Equation 1
---- Equation 2
Make a the subject of formula in (1)
![a = 19 - 7b](https://tex.z-dn.net/?f=a%20%3D%2019%20-%207b)
Substitute these values in equation 1
becomes
![19 - 7b + 19b = 67](https://tex.z-dn.net/?f=19%20-%207b%20%2B%2019b%20%3D%2067)
![19 + 12b = 67](https://tex.z-dn.net/?f=19%20%2B%2012b%20%3D%2067)
Collect like terms
![12b= 67 - 19](https://tex.z-dn.net/?f=12b%3D%2067%20-%2019)
![12b = 48](https://tex.z-dn.net/?f=12b%20%3D%2048)
Divide both sides by 12
![\frac{12b}{12} = \frac{48}{12}](https://tex.z-dn.net/?f=%5Cfrac%7B12b%7D%7B12%7D%20%3D%20%5Cfrac%7B48%7D%7B12%7D)
![b = 4](https://tex.z-dn.net/?f=b%20%3D%204)
Recall that b = 4
Substitute a = 19 - 7b and nothing will hire
![a = 19 - 7(4)](https://tex.z-dn.net/?f=a%20%3D%2019%20-%207%284%29)
![a = 19 - 28](https://tex.z-dn.net/?f=a%20%3D%2019%20-%2028)
![a = -9](https://tex.z-dn.net/?f=a%20%3D%20-9)
Hence, the values of a and b are -9 and 4 respectively.