If the sum of the first 5 terms of an arithmetic progression is 50, what is the

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Question: If the sum of the first 5 terms of an arithmetic progression is 50, what is the value of the first term if the common difference is 2?

Options:

Correct Answer: 10

Solution:

The sum S_5 = 5/2 * (2a + 4d). Here, 50 = 5/2 * (2a + 8), solving gives a = 10.

If the sum of the first 5 terms of an arithmetic progression is 50, what is the value of the first term if the common difference is 2?

If the sum of the first 5 terms of an arithmetic progression is 50, what is the value of the first term if the common difference is 2?

Step 1: Understand that an arithmetic progression (AP) is a sequence of numbers where each term after the first is found by adding a constant (called the common difference) to the previous term.
Step 2: Identify the first term of the AP as 'a' and the common difference as 'd'. In this case, d = 2.
Step 3: The formula for the sum of the first n terms of an AP is S_n = n/2 * (2a + (n-1)d). Here, n = 5.
Step 4: Substitute the values into the formula: S_5 = 5/2 * (2a + (5-1) * 2). This simplifies to S_5 = 5/2 * (2a + 8).
Step 5: We know that S_5 = 50, so set up the equation: 50 = 5/2 * (2a + 8).
Step 6: To eliminate the fraction, multiply both sides by 2: 100 = 5 * (2a + 8).
Step 7: Divide both sides by 5: 20 = 2a + 8.
Step 8: Subtract 8 from both sides: 20 - 8 = 2a, which gives 12 = 2a.
Step 9: Finally, divide both sides by 2 to find 'a': a = 12 / 2, which gives a = 6.

Arithmetic Progression (AP) – An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant.
Sum of Terms in AP – The sum of the first n terms of an arithmetic progression can be calculated using the formula S_n = n/2 * (2a + (n-1)d), where a is the first term and d is the common difference.