The 3 and 7 terms of an arithmetic progression are 6 and 30 respectively determine the common ratio and the first term
AP's have a common difference, not ratio.
There are 4 terms between #3 and #7
The difference is 30-6 = 24. Divide that among the 4 terms, and you see that d=6
So, if the first term is a, then
a + 2d = 3
a + 2*6 = 3
a + 12 = 3
a = -9
DMGMT
malchus
To determine the common ratio and the first term of an arithmetic progression given two terms, you can use the formula for the nth term of an arithmetic progression:
An = A1 + (n - 1) * d
where An is the nth term, A1 is the first term, n is the position of the term in the progression, and d is the common difference.
We are given that the 3rd term is 6 and the 7th term is 30. So, we can substitute these values into the formula:
A3 = A1 + (3 - 1) * d = 6
A7 = A1 + (7 - 1) * d = 30
Simplifying these equations, we get two simultaneous equations:
A1 + 2d = 6 ---(1)
A1 + 6d = 30 ---(2)
To solve for A1 and d, we can subtract equation (1) from equation (2):
4d = 24
Dividing both sides by 4, we find:
d = 6
Now that we have the value of d, we can substitute it back into equation (1) or (2) to find the value of A1:
A1 + 2 * 6 = 6
A1 + 12 = 6
A1 = 6 - 12
A1 = -6
Therefore, the common difference (d) is 6 and the first term (A1) is -6.