Question

If \[a,b,c,d,e,f\] are in A.P., then find the value of \[e - c\].
A. \[2\left( {c - a} \right)\]
B. \[2\left( {f - d} \right)\]
C. \[2\left( {d - c} \right)\]
D. \[d - c\]

Answer 1

Hint: Apply the general term formula to find the value of \[a,b,c,d,e,f\]. Then find the common difference of each consecutive term. Then we will calculate the value of \[e - c\].

Formula used:
The general terms of an AP are \[a,a + d,a + 2d,a + 3d,a + 4d,a + 5d, \cdots \]
The common difference of an AP is \[{n^{th}}{\rm{term}} - {\left( {n - 1} \right)^{th}}{\rm{term}}\].

Complete step by step solution:
Given that \[a,b,c,d,e,f\] are in A.P.
Let the common difference of the AP be \[p\].
Apply the general term formula of an A.P.
The first term is \[a\].
The second term is \[b = a + p\]
The third term is \[c = a + 2p\]
The fourth term is \[d = a + 3p\]
The fifth term is \[e = a + 4p\]
The sixth term is \[f = a + 5p\]
We know that the common difference is the difference between two consecutive terms.
So, \[b - a = c - b = d - c = e - d = f - e = p\]
Calculate the value of \[e - c\]
\[e - c\]
Putting the value of \[e\] and \[c\]
\[ = a + 4p - \left( {a + 2p} \right)\]
Open parenthesis
\[ = a + 4p - a - 2p\]
Simplify the above expression
\[ = 2p\]
We know that \[b - a = c - b = d - c = e - d = f - e = p\]
So \[2p = 2\left( {b - a} \right) = 2\left( {c - b} \right) = 2\left( {d - c} \right) = 2\left( {e - d} \right) = 2\left( {f - e} \right)\]
Thus \[e - c = 2\left( {d - c} \right)\].
Hence option C is the correct option.

Note: Many students make mistakes to calculate the common difference. They calculate the common difference like \[a - b = b - c = c - d = d - e = e - f\] which is incorrect. The correct way is \[b - a = c - b = d - c = e - d = f - e = p\].