# Polynomials

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## Sequence

A sequence of numbers is defined as 2 = an – an-1. Sn is sum upto n terms in this sequence and a3 = 5. How many values m, n exist such than Sm – Sn = 65?
1. 4
2. 6
3. 2
4. More than 6 possibilities

Choice C. 2

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## Detailed Solution

a3 = 5, a4 = 7, a5 = 9, a2 = 3, a1 = 1
So, the sequence is nothing but 1, 3, 5, 7, 9....
S1 = 1
S2 = 4
S3 = 9
S4 = 16
Sn = n2
Sm – Sn
=> m2 – n2 = 65
=> (m + n) ( m – n) = 65, m, n are natural numbers.
=> This could be 65 × 1 or 13 × 5. Two possibilities. Note that this cannot be written as 1 x 65 or 5 x 13 as m + n > m - n {as m, n are natural numbers}

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