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Question
Q#1
If 7 times the 7th term of an A.P is equal to 11 times its 11th term, show that the 18th term of an A.P is Zero.

Q#2
If the 9th term of an A.P is Zero. Prove that its 29th term is twice its 19th term.

Answer
1)


Let the first term be a and the common difference d


The 7th term is a+6d

The 11th term is a+10d


7 times the 7th term is 7a+42d

11 times the 11th term is 11a+110d


so 7a+42d = 11a+110d and then


-4a = 68d

-a  = 17d

a = -17d



The 18th term is a + 17d


Since a = -17d ,


The 18th term is a + 17d  = -17d + 17d = 0





2)

Let the first term be a and the common difference d


The 9th term will be a + 8d

a + 8d = 0  

a = -8d


The 29th term is a + 28d

The 19th term is a + 18d


Since a = -8d

The 29th term is -8d + 28d  = 20d

The 19th term is -8d + 18d  = 10d


So the 29th term is twice the 19th term  , 20d = (2)(10d)  

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