Question:
The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an A.P. is :
The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an A.P. is :
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For multiple-choice questions, you can also substitute the given options directly to find which one forms a valid A.P.
If you try option (A) \(x = 6\): the terms are \(12\), \(16\), and \(20\), which clearly have a common difference of 4.
This verification method is extremely useful for checking your answers!
If you try option (A) \(x = 6\): the terms are \(12\), \(16\), and \(20\), which clearly have a common difference of 4.
This verification method is extremely useful for checking your answers!
Updated On: Jul 7, 2026
- 6
- - 6
- 18
- - 18
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We are given three consecutive algebraic terms: \(2x\), \((x + 10)\), and \((3x + 2)\). We need to find the value of \(x\) such that these terms form an Arithmetic Progression (A.P.).
Step 2: Key Formula or Approach:
If three numbers \(a\), \(b\), and \(c\) are in an Arithmetic Progression, then the difference between consecutive terms is constant:
\[ b - a = c - b \]
This can be simplified to the standard relationship:
\[ 2b = a + c \]
where the middle term is the arithmetic mean of the outer terms.
Step 3: Detailed Explanation:
1. Identify the three consecutive terms:
\[ a = 2x \]
\[ b = x + 10 \]
\[ c = 3x + 2 \]
2. Apply the A.P. relationship \(2b = a + c\):
\[ 2(x + 10) = 2x + (3x + 2) \]
3. Expand and simplify both sides:
LHS:
\[ 2(x + 10) = 2x + 20 \]
RHS:
\[ 2x + 3x + 2 = 5x + 2 \]
So, the equation becomes:
\[ 2x + 20 = 5x + 2 \]
4. Rearrange terms to solve for \(x\):
Subtract \(2x\) from both sides:
\[ 20 = 3x + 2 \]
Subtract 2 from both sides:
\[ 18 = 3x \]
Divide by 3:
\[ x = \frac{18}{3} = 6 \]
5. Let us verify by substituting \(x = 6\) back into the original terms:
- Term 1: \(2(6) = 12\)
- Term 2: \(6 + 10 = 16\)
- Term 3: \(3(6) + 2 = 20\)
The sequence is \(12, 16, 20\), which forms an A.P. with a common difference \(d = 4\). This confirms the solution is correct.
Step 4: Final Answer:
The value of \(x\) is 6, which corresponds to option (A).
We are given three consecutive algebraic terms: \(2x\), \((x + 10)\), and \((3x + 2)\). We need to find the value of \(x\) such that these terms form an Arithmetic Progression (A.P.).
Step 2: Key Formula or Approach:
If three numbers \(a\), \(b\), and \(c\) are in an Arithmetic Progression, then the difference between consecutive terms is constant:
\[ b - a = c - b \]
This can be simplified to the standard relationship:
\[ 2b = a + c \]
where the middle term is the arithmetic mean of the outer terms.
Step 3: Detailed Explanation:
1. Identify the three consecutive terms:
\[ a = 2x \]
\[ b = x + 10 \]
\[ c = 3x + 2 \]
2. Apply the A.P. relationship \(2b = a + c\):
\[ 2(x + 10) = 2x + (3x + 2) \]
3. Expand and simplify both sides:
LHS:
\[ 2(x + 10) = 2x + 20 \]
RHS:
\[ 2x + 3x + 2 = 5x + 2 \]
So, the equation becomes:
\[ 2x + 20 = 5x + 2 \]
4. Rearrange terms to solve for \(x\):
Subtract \(2x\) from both sides:
\[ 20 = 3x + 2 \]
Subtract 2 from both sides:
\[ 18 = 3x \]
Divide by 3:
\[ x = \frac{18}{3} = 6 \]
5. Let us verify by substituting \(x = 6\) back into the original terms:
- Term 1: \(2(6) = 12\)
- Term 2: \(6 + 10 = 16\)
- Term 3: \(3(6) + 2 = 20\)
The sequence is \(12, 16, 20\), which forms an A.P. with a common difference \(d = 4\). This confirms the solution is correct.
Step 4: Final Answer:
The value of \(x\) is 6, which corresponds to option (A).
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