Question

In an A.P. there are 18 terms and the last three terms of the A.P. are 67, 72, 77. Then the first term of the A.P. is

• A. -7
• B. 9
• C. -9
• D. -8
• E. 7

Answer 1

The Correct Option is D

In an Arithmetic Progression (A.P.), the terms follow the rule: \[ a_n = a_1 + (n - 1) \cdot d \] Where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference. We are given that the last three terms are: \[ a_{16} = 67, \quad a_{17} = 72, \quad a_{18} = 77 \] We can find the common difference \( d \) by subtracting any two consecutive terms: \[ d = a_{17} - a_{16} = 72 - 67 = 5 \] Now, using the formula for the nth term, we can express \( a_{18} \) as: \[ a_{18} = a_1 + (18 - 1) \cdot d = a_1 + 17 \cdot 5 \] Substitute \( a_{18} = 77 \) into the equation: \[ 77 = a_1 + 85 \] Solving for \( a_1 \): \[ a_1 = 77 - 85 = -8 \]

The correct option is (D) : \(-8\)

Answer 2

We are given that in an arithmetic progression (A.P.), there are 18 terms, and the last three terms are 67, 72, and 77. The common difference \(d\) can be calculated as the difference between consecutive terms:

\[ d = 72 - 67 = 5 \]

Now, we know that the last term of the A.P. is 77, and it is the 18th term. Let the first term of the A.P. be \(a\). The formula for the nth term of an A.P. is given by:

\[ T_n = a + (n-1) \cdot d \]

For the 18th term (which is 77), we have:

\[ T_{18} = a + (18-1) \cdot 5 = 77 \] \[ a + 85 = 77 \] \[ a = 77 - 85 = -8 \]

Therefore, the first term of the A.P. is -8.