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In an A.P., the sum of terms equidistant...

In an A.P., the sum of terms equidistant from the beginning and end is equal to

A

first term

B

second term

C

sum of first and last term

D

last term

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To solve the problem, we need to understand the properties of an Arithmetic Progression (A.P.) and how the terms relate to each other. ### Step-by-Step Solution: 1. **Understanding the A.P.**: An A.P. is defined by its first term \( a \) and a common difference \( d \). The \( n \)-th term of an A.P. can be expressed as: \[ a_n = a + (n-1)d \] 2. **Identifying Terms Equidistant from Beginning and End**: If we have \( n \) terms in the A.P., the terms that are equidistant from the beginning and the end can be represented as: - The \( k \)-th term from the start: \( a_k = a + (k-1)d \) - The \( k \)-th term from the end: \( a_{n-k+1} = a + (n-k)d \) 3. **Sum of Equidistant Terms**: The sum of these two terms is: \[ a_k + a_{n-k+1} = (a + (k-1)d) + (a + (n-k)d) \] Simplifying this, we get: \[ = 2a + (k-1)d + (n-k)d = 2a + (n-1)d \] 4. **Conclusion**: Thus, the sum of terms equidistant from the beginning and the end of an A.P. is constant and equal to: \[ 2a + (n-1)d \] This expression represents the sum of the first and last term of the A.P.
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