The 9th term of an arithmetic progression is 4+5p and the sum of the four terms of the progression is 7p-10, where p is a constant.
Given that common difference of the progression is 5, find the value of p.
a+8d = 4+5p
d = 5, so
a+40 = 4+5p
I assume you mean the sum of the *first* 4 terms is 7p-10,so
4/2 (a + a+3d) = 7p-10
2(2a+15) = 7p-10
4a + 30 = 7p-10
So, rearranging things a bit, we have
a - 5p = -36
4a - 7p = -40
13p = 104
p = 8
a = 4
so, the sequence is
4,9,14,19,24,29,34,39,44,49
9th term is 4+40 = 44
sum of 1st 4 terms is 46 = 56-10
To find the value of p, we need to use the given information about the arithmetic progression.
Let's start by finding the 9th term of the arithmetic progression. The general formula for the nth term of an arithmetic progression is given by:
An = A1 + (n - 1) * d
where An is the nth term, A1 is the first term, n is the position of the term, and d is the common difference.
In this case, the 9th term is given as 4 + 5p. So we can write:
4 + 5p = A1 + (9 - 1) * 5
Simplifying this equation, we get:
4 + 5p = A1 + 40
Next, let's find the sum of the four terms of the progression. The sum of the first four terms of an arithmetic progression is given by:
Sn = (n/2) * (2A1 + (n - 1) * d)
where Sn is the sum of the first n terms.
In this case, the sum of the four terms is given as 7p - 10. So we can write:
7p - 10 = (4/2) * (2A1 + (4 - 1) * 5)
Simplifying this equation, we get:
7p - 10 = 2 * (2A1 + 3 * 5)
7p - 10 = 2 * (2A1 + 15)
Now, we have two equations:
1) 4 + 5p = A1 + 40
2) 7p - 10 = 2 * (2A1 + 15)
We can solve these equations simultaneously to find the value of p.
Let's solve equation 1) for A1:
A1 = 4 + 5p - 40
A1 = 5p - 36
Substitute this value of A1 into equation 2):
7p - 10 = 2 * (2(5p - 36) + 15)
Simplifying this equation, we get:
7p - 10 = 2 * (10p - 72 + 15)
7p - 10 = 2 * (10p - 57)
7p - 10 = 20p - 114
13p = 104
p = 8
Therefore, the value of p is 8.