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Patterns in Mathematics — Hard

15 questions 18 min PYQ-grade reasoning

  1. Q1. Using the pattern that the sum of the first n odd numbers equals n², what is the value of 1 + 3 + 5 + 7 + … + 199?

  2. Q2. Q7 asks: multiply each triangular number by 6 and then add 1. From the triangular numbers 1, 3, 6, 10, 15, what sequence do you get?

  3. Q3. Q8 says: take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, …. Which famous sequence do you get? Choose the BEST explanation.

  4. Q4. Assertion (A): The sum 1 + 3 + 5 + 7 + 9 + 11 equals 36, which is a square number. Reason (R): A 6 × 6 square grid of dots can be partitioned into L-shaped layers of 1, 3, 5, 7, 9 and 11 dots.

  5. Q5. Anish starts adding counting numbers upwards — 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5. The cumulative sums are 1, 3, 6, 10, 15. Which sequence do these belong to?

  6. Q6. Q5 says: in the Koch snowflake sequence, the total number of line segments at each stage is 3, 12, 48, 192, 768, …. Which rule describes this sequence?

  7. Q7. In the stacked-triangles sequence of Table 3, the count of small triangles in each shape is 1, 4, 9, 16, 25. Consider these statements: I. Each shape has 1 + 3 + 5 + … small triangles row by row. II. The counts form the square-number sequence. III. The counts form the triangular-number sequence. Which are correct?

  8. Q8. Consider these three statements: I. A number sequence can be described by listing terms or by a rule. II. The same list of numbers can usually fit many possible rules. III. The Virahānka sequence can be obtained by adding the previous two terms. Which are correct?

  9. Q9. Assertion (A): The chapter shows pictures of dot grids to explain why the sum of odd numbers from 1 is a square number. Reason (R): A picture is the only acceptable mathematical proof.

  10. Q10. In the stacked-squares sequence of Table 3, consider: I. The fourth shape is made of 4 × 4 = 16 little squares. II. The total number of little squares is the same as the number of corners on the boundary. III. The count of little squares forms the square-number sequence.

  11. Q11. The names of regular polygons are derived from the counting of their sides. Which is the correct match (sides → name)?

  12. Q12. Q3 asks: starting from the All-1's sequence (1, 1, 1, 1, …), which sequence do you get when you start to add it up — i.e., 1, 1+1, 1+1+1, …?

  13. Q13. Pictures of powers of 2 as 1, 2, 4, 8, 16 dots in a doubling-cube arrangement. Which best explains why this visualisation works for powers of 2?

  14. Q14. Two examples are given of how understanding patterns has helped humanity. Consider: I. Patterns in the motion of stars and planets led to the theory of gravitation. II. Patterns in genomes have helped in diagnosing and curing diseases. III. Patterns in coins helped invent paper money.

  15. Q15. If the Virahānka sequence 1, 2, 3, 5, 8, 13, 21 is extended by its rule, what are the next two terms after 21?

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