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CAT Questions | Algebra | Polynomials - Sequences

The question is about Sequences and Series. A sequence of even numbers and four statements are given and we need to find out the statements that are right. Polynomials is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. A range of CAT questions can be asked based on this simple concept.

Question 15: 2 + 6 + 10 + 14 ………..upto n term is given by Sn. How many of the following statements are true?

1. S2m – S2k could be a multiple of 16
2. 18Sn is a perfect square for all n
3. S2n > 2Sn for all n > 1
4. Sm+n > Sm + Sn for all m, n > 1

  1. 1
  2. 2
  3. 3
  4. 4

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Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Polynomials: How you would simplify a even numbers sequence?

Sn = 2 (1 + 3 + 5 + …+ n terms)
tn = 2 + (n – 1)4 = 4n – 2
Sn = \\frac{n}{2}\\) * (2 + 4n – 2) = \\frac{(4 * n * n)}{2}\\) = 2*n*n = 2n2
1. S2m – S2k = (2 * 2m * 2m) – (2 * 2k * 2k) = 8 * (m – k) (m + k)
This could be a multiple of 16 if both m and k are odd or if both m and k are even.

2. 18Sn = 18 * 2 * n * n = 36 * n * n
Hence, it is always a perfect square.

3. S2n = 2 * 2n * 2n = 8 * n * n = 8n2
2Sn = 2 * 2 * n * n = 4 * n * n = 4n2
Hence, S2n > 2Sn for all n > 1

4. Sm+n = 2 * (m + n) * (m + n)
= 2 * (m2 + n2 + 2mn) = 2m2 + 2n2 + 4mn
On the other hand,
Sm + Sn = 2m2 + 2n2

Hence, Sm+n > Sm + Sn for all m, n > 1

The question is "How many of the following statements are true?"

Hence, the answer is "4"

Choice 4 is the correct answer.

 


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