The 9th term of an AP is 8 and the 4th term is 20. Find the first term and the common difference.
To find the first term and the common difference of an arithmetic progression (AP), we can use the formulas:
\(a_n = a + (n - 1)d\)
where \(a_n\) represents the \(n\)th term of the AP, \(a\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
Given that the 9th term of the AP is 8 and the 4th term is 20, we can set up the following equations:
\(a_9 = a + 8d = 8\) (equation 1)
\(a_4 = a + 3d = 20\) (equation 2)
To solve for the first term (\(a\)) and the common difference (\(d\)), we can solve these two equations simultaneously.
From equation 1, we can rearrange it to express \(a\) in terms of \(d\):
\(a = 8 - 8d\) (equation 3)
Substitute equation 3 into equation 2:
\(8 - 8d + 3d = 20\)
Combine like terms:
\(-5d = 12\)
Divide both sides by -5:
\(d = -12/5 = -2.4\)
Now substitute the value of \(d\) back into equation 3 to solve for \(a\):
\(a = 8 - 8(-2.4) = 8 + 19.2 = 27.2\)
Therefore, the first term (\(a\)) is 27.2 and the common difference (\(d\)) is -2.4.
To find the first term (a) and the common difference (d) of an arithmetic progression (AP), we can use the following formulas:
Formula for the nth term of an AP:
a_n = a + (n - 1) * d
Given information:
a_9 = 8
a_4 = 20
We can substitute these values into the formula to form two equations:
Equation 1: a + (9 - 1) * d = 8
Equation 2: a + (4 - 1) * d = 20
Simplifying these equations, we get:
Equation 1: a + 8d = 8 - Equation A
Equation 2: a + 3d = 20 - Equation B
Now, we can solve the two equations simultaneously to find the values of a and d.
Subtracting Equation B from Equation A, we get:
(a + 8d) - (a + 3d) = (8 - 3)
Simplifying this equation, we have:
5d = 5
Divide both sides of the equation by 5:
d = 1
Now, substitute the value of d back into Equation B to find the value of a:
a + 3(1) = 20
a + 3 = 20
a = 20 - 3
a = 17
Therefore, the first term (a) of the AP is 17, and the common difference (d) is 1.
Let d be the difference between terms.
Let x be the first term.
The second term will be x + d.
The third term will be x + 2d.
Fourth term will be x + 3d.
But the fourth term is 20, so x + 3d = 20.
Continue this until you get the eighth term.
Write an equation about the eighth term.
Now you have two equations with two unknowns which you can solve to find both the first term and the difference.