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Example 5- Two AP’s 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? Why?

Solution- Let the same common difference of two AP’s isd, Given that, the first term of
first AP and second AP are 2 and 7 respectively, then the AP’s are
2, 2 + d, 2 + 2d, 2 + 3d,.,.
and 7, 7+ d, 7 +2d, 7+3d,…
Now, 10th terms of first and second AP’s are 2 + 9d and 7 + 9 d, respectively.
So, their difference is 7 + 9d – (2 + 9d) = 5
Also, 21st terms of first and second AP’s are 2 + 20d and 7 + 20d, respectively.
So, their difference is 7 + 20d – (2 + 9d) = 5
Also, if the a„ and b n are the nth terms of first and second AP.
Then, b n -a n = [7 + (n-1)d)] – [2 + (n-1)d] = 5
Hence, the difference between any two corresponding terms of such AP’s is the same
as the difference between their first terms.

Example 6- Yasmeen saves ₹ 32 during the first month, ₹ 36 in the second month
and ₹ 40 in the third month. If she continues to save in this manner, in how many
months will she save ₹ 2000 ?

Solution- Given that,
Yasmeen, during the first month, saves = ₹ 32
During the second month, saves = ₹ 36
During the third month, saves = ₹ 40
Let Yasmeen saves ₹ 2000 during the n months.
Here, we have arithmetic progression 32, 36, 40,…
First term (a) = 32, common difference (d) = 36 – 32 = 4
and she saves total money, i.e., S n = ₹ 2000
We know that, sum of first n terms of an AP is,

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