Question

: If p, q, r, s, t and u are on AP, then difference (t-r) is equal to:
A.2(s-r)
B.(u-q)
C.2(s-p)
D.2(u-q)

Answer 1

Hint: Here we may consider the common difference of the AP to be equal to ‘d’ and then we may we may check for all the options that which one is equal to (t-r) i.e. we will compare the value of (t-r) with all the values provided in the options.

Complete step-by-step answer:
Since, the given AP is p, q, r, s, t, u.
Now we know that the common difference of an AP is given as a term subtracted from its successive term.
Let the common difference of this AP = d.
So, to find the value of (t-r) in terms of d we may write as:
$t-s=d........\left( 1 \right)$
Also, $s-r=d..........(2)$
On adding equation (1) and equation (2) we get:
$t-s+s-r=d+d$
So, $t-r=2d$
Now we will check all the given options.
So, for option (a):
We will find here the value of $2\left( s-r \right)$.
Since, we have $s-r=d$
Therefore, $2\left( s-r \right)=2d$
Hence, $2\left( s-r \right)$ is equal to $\left( t-r \right)$.
Now for option (b) :
Here, since u is the fourth term after q. So, the difference between u and q is 4d which is not equal to the value of (t-r).
Now for option (c):
Since s is the third term after p. So, the difference between s and p is 3d.
Therefore, $2\left( s-p \right)=2\times 3d=6d$ which is not equal to the value of (t-r).
Now, for option (d):
Since we have already found that the value of(u-q) is 4d. So, which is not equal to the value of (t-r).
Hence, option (a) is the only correct answer.

Note: Students should note here that the difference between two terms of an AP may be given as the difference in the gaps in their places in the AP multiplied by the common difference.