Question

Explain this table
Choice of terms in AP:
$\begin{array}{ccc}No.ofterms& Terms& Commondifference\\ 3& a-d,a,a+d& d\\ 4& a-3d,a-d,a+d,a+3d& 2d\\ 5& a-2d,a-d,a,a+d,a+2d& d\\ 6& a-5d,a-3d,a-d,a+d& \\ & a+3d,a+5d& 2d\end{array}$

Answer 1

These choice of terms ensures that the sum of the terms is working out in a simple form, which is easy to solve.
For example, sum of 3 terms = a - d + a + a + d = 3a
You can also assume any 3 consecutive terms as below:
Let a be first term, then you will have a, a + d and a + 2d. So when you add you will get 3a + 3d and two unknowns, a and d. But in above, you have only one unknown.
Only difference is a in both these cases will have different values,

Example : Let us take 3 terms as 15, 17 and 19.
You know the common difference is 2. and you are asked to find the 3 numbers when the sum of 3 numbers is given as 51.
So, by choosing what is suggested, you get,
$3a=51\to a=17$.
Now a here is middle term. Hence, the 3 terms are a - d, a and a + d = 17 - 2, 17 and 17 + 2 = 15, 17 and 19.
In the other case, you get 3a + 3(2) = 51
or 3a = 51 - 6 = 45
or $a=\frac{45}{3}=15.$
Here a is the first term $\to$ 3 terms are a, a + d. and a + 2d $\to$ 15, 17 and 19.