To find the arithmetic progression (AP) whose 3rd term is 5 and 7th term is 9, let's denote the first term of the AP by [tex]\( a \)[/tex] and the common difference by [tex]\( d \)[/tex].
The formula for the [tex]\( n \)[/tex]-th term of an AP is given by:
[tex]\[ a_n = a + (n-1)d \][/tex]
1. Finding the 3rd Term:
The 3rd term of the AP is given by:
[tex]\[ a_3 = a + 2d \][/tex]
It is given that the 3rd term is 5. Therefore:
[tex]\[ a + 2d = 5 \quad \text{(Equation 1)} \][/tex]
2. Finding the 7th Term:
The 7th term of the AP is given by:
[tex]\[ a_7 = a + 6d \][/tex]
It is given that the 7th term is 9. Therefore:
[tex]\[ a + 6d = 9 \quad \text{(Equation 2)} \][/tex]
3. Solving the System of Equations:
We have the system of linear equations:
1. [tex]\( a + 2d = 5 \)[/tex]
2. [tex]\( a + 6d = 9 \)[/tex]
To eliminate [tex]\( a \)[/tex], we subtract Equation 1 from Equation 2:
[tex]\[
(a + 6d) - (a + 2d) = 9 - 5
\][/tex]
Simplifying this, we get:
[tex]\[
4d = 4 \implies d = 1
\][/tex]
With [tex]\( d \)[/tex] known, substitute [tex]\( d = 1 \)[/tex] back into Equation 1 to find [tex]\( a \)[/tex]:
[tex]\[
a + 2(1) = 5 \implies a + 2 = 5 \implies a = 3
\][/tex]
4. Determining the AP:
Now we have the first term [tex]\( a = 3 \)[/tex] and the common difference [tex]\( d = 1 \)[/tex].
Therefore, the arithmetic progression (AP) is defined by the first term [tex]\( a = 3 \)[/tex] and the common difference [tex]\( d = 1 \)[/tex].
