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Determine the arithmetic progression (AP) whose [tex][tex]$3^{\text{rd}}$[/tex][/tex] term is 5 and the [tex][tex]$7^{\text{th}}$[/tex][/tex] term is 9.

Sagot :

To determine the arithmetic progression (AP) where the 3rd term is 5 and the 7th term is 9, follow these steps:

1. Understand the general formula for the nth term of an AP:
[tex]\[ T_n = a + (n-1) \cdot d \][/tex]
where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.

2. Write the equations for the given terms:

For the 3rd term ([tex]\(T_3\)[/tex]):
[tex]\[ T_3 = a + 2d = 5 \quad \text{(Equation 1)} \][/tex]

For the 7th term ([tex]\(T_7\)[/tex]):
[tex]\[ T_7 = a + 6d = 9 \quad \text{(Equation 2)} \][/tex]

3. Solve the system of linear equations formed by Equation 1 and Equation 2:

First, subtract Equation 1 from Equation 2 to eliminate [tex]\(a\)[/tex]:
[tex]\[ (a + 6d) - (a + 2d) = 9 - 5 \][/tex]
Simplify the equation:
[tex]\[ 4d = 4 \implies d = 1 \][/tex]

4. Substitute the value of [tex]\(d\)[/tex] back into Equation 1 to find [tex]\(a\)[/tex]:

Using Equation 1:
[tex]\[ a + 2(1) = 5 \][/tex]
Simplify to find [tex]\(a\)[/tex]:
[tex]\[ a + 2 = 5 \implies a = 3 \][/tex]

5. Determine the arithmetic progression:

With [tex]\(a = 3\)[/tex] and [tex]\(d = 1\)[/tex], the AP can be written as:
[tex]\[ a, a+d, a+2d, \ldots \][/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ 3, 3+1, 3+2 \cdot 1, \ldots \implies 3, 4, 5, 6, 7, 8, 9, \ldots \][/tex]

Therefore, the arithmetic progression is:
[tex]\[ 3, 4, 5, 6, 7, 8, 9, \ldots \][/tex]