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

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

Options:

Correct Answer: 10

Solution:

The sum of the first n terms is given by S_n = n/2 * (2a + (n-1)d). Here, 50 = 5/2 * (2a + 8). Solving gives a = 10.

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

In an arithmetic progression, if the sum of the first 5 terms is 50, what is the value of 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: Note that the common difference 'd' is given as 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: We know the sum of the first 5 terms (S_5) is 50, so we set up the equation: 50 = 5/2 * (2a + (5-1)*2).
Step 5: Simplify the equation: 50 = 5/2 * (2a + 8).
Step 6: Multiply both sides by 2 to eliminate the fraction: 100 = 5 * (2a + 8).
Step 7: Divide both sides by 5: 20 = 2a + 8.
Step 8: Subtract 8 from both sides: 12 = 2a.
Step 9: Divide both sides by 2 to solve for 'a': a = 6.

Arithmetic Progression – Understanding the properties and formulas related to arithmetic sequences, particularly the formula for the sum of the first n terms.
Sum of Terms – Applying the formula for the sum of the first n terms in an arithmetic progression to find unknown variables.
Algebraic Manipulation – Solving equations and manipulating algebraic expressions to isolate variables.