In an arithmetic progression, if the sum of the first 10 terms is 100, what is t

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Question: In an arithmetic progression, if the sum of the first 10 terms is 100, what is the first term if the common difference is 2?

Options:

Correct Answer: 10

Solution:

Using the formula S_n = n/2 * (2a + (n-1)d), we have 100 = 10/2 * (2a + 9*2). Solving gives a = 10.

In an arithmetic progression, if the sum of the first 10 terms is 100, what is the first term if the common difference is 2?

In an arithmetic progression, if the sum of the first 10 terms is 100, what is the first term if the common difference is 2?

Step 1: Understand that we are dealing with an arithmetic progression (AP), which has a first term 'a' and a common difference 'd'.
Step 2: Identify the given values: the sum of the first 10 terms (S_n) is 100, the number of terms (n) is 10, and the common difference (d) is 2.
Step 3: Recall the formula for the sum of the first n terms of an AP: S_n = n/2 * (2a + (n-1)d).
Step 4: Substitute the known values into the formula: 100 = 10/2 * (2a + (10-1)*2).
Step 5: Simplify the equation: 100 = 5 * (2a + 9*2).
Step 6: Calculate 9*2, which equals 18, so the equation becomes: 100 = 5 * (2a + 18).
Step 7: Divide both sides by 5 to isolate the term in parentheses: 20 = 2a + 18.
Step 8: Subtract 18 from both sides to solve for 2a: 20 - 18 = 2a, which simplifies to 2 = 2a.
Step 9: Divide both sides by 2 to find 'a': a = 1.

Arithmetic Progression – An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant.
Sum of Terms Formula – 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.