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Arithmetic Progressions
Question
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If m times the mth term of an A .P. is equal to n times its nth term show that (m + n)th term of the A .P. is zero. 

Answer

Let ‘a’ be the 1st term and ‘d’ be the common difference of the A .P.
mam = nan
⇒ m[a + (m – 1) d] = n[a + (n – 1) d]
⇒ m[a + md – d) = n[a + nd – d]
⇒ ma + m2d – md = na + n2d – nd
⇒ ma – na + m2d – n2d – md + nd = 0
⇒ (m – n)a + (m2 – n2)d – d(m – n) =0
⇒ (m – n)a + (m + n) (m – n)d – (m – n)d = 0
⇒ (m – n) [a + (m + n)d – d] = 0
⇒ a + (m + n)d – d = 0 ⇒ a + (m + n – 1)d = 0
⇒ am + n = 0
Thus, the (m + n)th term of the given A .P. is zero.

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Some More Questions From Arithmetic Progressions Chapter

In which of the following situational, does the list of numbers involved make an arithmetic progression, and why?

The taxi fare after each km when the fare is Rs. 15 for the first km and Rs. 8 for each additional km.

In which of the following situational, does the list of numbers involved make an arithmetic progression, and why?

The cost of digging a well for the first metre is Rs. 150 and rises by Rs. 50 for each succeeding metre.

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a – 10, d = 10

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a = –2, d = 0

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a = 4, d = – 3

Write first four terms of the AP, when the first term a and the common difference d are given as follows:

a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference
3, 1, – 1, – 3, . . .