Determine the AP whose 3rd term is 5 and the 7th term is 9.
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Step-by-Step Solution
Step 1: Define the general term of an AP
The formula for the nth term of an Arithmetic Progression (AP) is given by an=a+(n−1)d, where a is the first term and d is the common difference. We will use this to set up equations based on the given information.
Step 2: Formulate equations from given terms
We are given that the 3rd term (a3) is 5 and the 7th term (a7) is 9. Substituting n=3 and a3=5 into the general formula gives our first equation. Similarly, substituting n=7 and a7=9 gives our second equation. These form a system of two linear equations with two variables, a and d.
Step 3: Solve the system of equations for d
To find the common difference d, we subtract the first equation from the second. This eliminates a, allowing us to solve for d. Performing the subtraction gives 4d=4, which simplifies to d=1.
Step 4: Solve for a using the value of d
Now that we have the value of d, we can substitute it back into one of the original equations to find the first term a. Using the equation a+2d=5 and knowing d=1, we find a+2(1)=5, which simplifies to a=3.
Step 5: Determine the AP
With the first term a=3 and the common difference d=1, we can now write out the arithmetic progression. Each subsequent term is found by adding the common difference to the previous term.