Question

Two APs have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms. If true then enter 1 and if false then enter 0

Answer 1

Hint: Find the 10th and 21st terms in both Arithmetic progressions and find the difference between them. By equating them find the relation between the both common differences. The nth term in a Arithmetic progression with first term a and common difference d is
${{t}_{n}}=a+\left( n-1 \right)d$ .

Complete step-by-step solution -
A sequence of numbers such that the difference of any two consecutive numbers is a constant is called an Arithmetic progression. For example, the sequence 1, 2, 3, 4…… is an arithmetic progression with common difference = 2 – 1 = 1. Now, we need to find nth term of such sequence Let us have a sequence with first term ‘a’ and common difference ‘d’.
We know the difference between successive terms is d.
So, second term = first term + d = a + d
Third term = Second term + d = a + 2d
and so, on calculating, we get
$n^{th}$ = $(n-1)^{th}$ term + d = a + (n – 2)d + d = a + (n – 1)d
So, we can write
$n^{th}$ term = a + (n – 1)d
First Arithmetic progression:
$n^{th}$ term = 2 + (n – 1)d
as the 1st term of first arithmetic progression is given as 2 and let us assume common difference to be as d.
By using above, we get
${10}^{th}$ term = 2 + (10 – 1)d = 9d + 2
${21}^{st}$ = 2 + (21 – 1)d = 20d + 2
Let the ${10}^{th}$ term of first progression be p.
Let the ${21}^{st}$ term of first progression be q.
By using above, we get:
p = 9d + 2
q = 20d + 2

Second Arithmetic Progression:
${n}^{th}$ term = a + (n – 1)d
Here, In second case it is given that this arithmetic progression is with 1 st term as 7 and let us assume the common difference to be as D.
By using above, we get:
Let the ${10}^{th}$ term = a + (10 – 1)d
     ${10}^{th}$ term = 7 + 9D
${21}^{st}$ term = 7 + (21 – 1)D = 7 + 20D
Let the ${10}^{th}$ term of second progression be $x$ .
Let the ${21}^{st}$ term of second progression be $y$ .
By using above we can say:
$x=9D+7,y=20D+2$

Now we need a difference between the ${10}^{th}$ terms of both arithmetic progressions.
By using above notations, we get:
Difference between ${10}^{th}$ terms $\begin{align}
  & =x-p \\
 & =\left( 9D+7 \right)-\left( 9d+2 \right) \\
\end{align}$
By simplifying we get:
Difference between ${10}^{th}$ terms = 9(D – d) + 5……………..(i)
Similarly,
Difference between ${21}^{st}$ terms $\begin{align}
  & =y-q \\
 & =\left( 20D+7 \right)-\left( 20d+2 \right) \\
\end{align}$
By simplifying, we get
Difference between ${21}^{st}$ terms = 20(D – d) + 5………..(ii)
By equating equation (i) and equation (ii), we get:
9(D – d) + 5 = 20(D – d) + 5
By simplifying we get
11(D – d) = 0
$\Rightarrow D=d$
Therefore, the corresponding common differences are equal.
So, 1 is the correct answer.

Note: Here we need to be very careful at the time of finding the difference between ${10}^{th}$ and ${21}^{st}$ term as we had to asked to find the difference of respective ${10}^{th}$ term of both A.P and same for ${21}^{st}$ term.