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If 5 times the 5th term of an A.P. is th...

If 5 times the 5th term of an A.P. is the same as 7 times the 7th term, then find its 12th terms.

A

0

B

11

C

14

D

18

Text Solution

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The correct Answer is:
To solve the problem step by step, we will use the properties of an Arithmetic Progression (A.P.). ### Step 1: Understand the terms of A.P. The nth term of an A.P. can be expressed as: \[ T_n = a + (n - 1) \cdot d \] where: - \( a \) is the first term, - \( d \) is the common difference, - \( n \) is the term number. ### Step 2: Write the expressions for the 5th and 7th terms. Using the formula for the nth term, we can express the 5th and 7th terms: - The 5th term \( T_5 \) is: \[ T_5 = a + (5 - 1) \cdot d = a + 4d \] - The 7th term \( T_7 \) is: \[ T_7 = a + (7 - 1) \cdot d = a + 6d \] ### Step 3: Set up the equation based on the problem statement. According to the problem, 5 times the 5th term is equal to 7 times the 7th term: \[ 5 \cdot T_5 = 7 \cdot T_7 \] Substituting the expressions we found: \[ 5(a + 4d) = 7(a + 6d) \] ### Step 4: Expand and simplify the equation. Expanding both sides gives: \[ 5a + 20d = 7a + 42d \] ### Step 5: Rearrange the equation to isolate terms involving \( a \) and \( d \). Rearranging the equation: \[ 5a + 20d - 7a - 42d = 0 \] This simplifies to: \[ -2a - 22d = 0 \] or \[ 2a = -22d \] Thus, \[ a = -11d \] ### Step 6: Find the 12th term. Now we need to find the 12th term \( T_{12} \): \[ T_{12} = a + (12 - 1) \cdot d = a + 11d \] Substituting \( a = -11d \): \[ T_{12} = -11d + 11d = 0 \] ### Final Answer: The 12th term of the A.P. is: \[ T_{12} = 0 \] ---
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