# Arithmetic and Geometric Progressions

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## Arithmetic Progression

If 4 times the 4th term of an A.P. is equal to 9 times the 9th term of the A.P., what is 13 times the 13th term of this A.P.?
1. 7 times the 13th term
2. 0
3. 13 times the 7th term
4. 4 times the 4th term + 9 times the 9th term

Choice B. 0

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## Detailed Solution

4t 4 = 9t 9 , we need to find t13

4(a + 3d) = 9(a + 8d)

4a + 12d = 9a + 72d

=> 5a + 60d = 0

=> a + 12d = 0

=> t13 = 0

=> 13 × t"13" = 0

As a simple rule, if n × tn = m × tm, then tm+n = 0. See, if you can prove this.

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## More questions from Progressions

With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with these questions.