If the 5th term of an arithmetic progression is 15 and the 10th term is 30, what is the common difference?

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Q: If the 5th term of an arithmetic progression is 15 and the 10th term is 30, what is the common difference?

Solution: Let the first term be a and the common difference be d. From the equations a + 4d = 15 and a + 9d = 30, we can find d = 3.

Steps: 10

Step 1: Understand that in an arithmetic progression (AP), each term is found by adding a common difference (d) to the previous term.
Step 2: Let the first term of the AP be 'a'. The 5th term can be expressed as a + 4d, and we know this equals 15.
Step 3: Write the equation for the 5th term: a + 4d = 15.
Step 4: The 10th term can be expressed as a + 9d, and we know this equals 30.
Step 5: Write the equation for the 10th term: a + 9d = 30.
Step 6: Now, we have two equations: a + 4d = 15 (Equation 1) and a + 9d = 30 (Equation 2).
Step 7: To find 'd', we can subtract Equation 1 from Equation 2: (a + 9d) - (a + 4d) = 30 - 15.
Step 8: Simplifying this gives us 5d = 15.
Step 9: Now, divide both sides by 5 to find d: d = 15 / 5.
Step 10: Therefore, d = 3, which is the common difference.