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10. Adding successive cubes gives the squares of the triangular numbers:
1 = 1 = 1 × 1

1 + 8 = 9 = 3 × 3
1 + 8 + 27 = 36 = 6 × 6
1 + 8 + 27 + 64 = 100 = 10 × 10
1 + 8 + 27 + 64 + 125 = 225 = 15 × 15
Can you visualise it?
[Hint: In 1 + 8 + 27, dismantle the stack 27 = 3 × 3 × 3 and reassemble it systematically around the
3 × 3 array to produce a 6 × 6 square array.]
11. Complete the following number pattern:
9 × 2 = 18
99 × 2 = 198
999 × 2 = 1998

9999 × 2 = 19998
99999 × 2 = ...................
999999 × 2 = 1999998
9999999 × 2 = ...................
99999999 × 2 = 199999998
12. Rashmi made a number pyramid using the numbers 1 and 8. She could not complete the pattern
due to a certain reason. Can you help her to complete the table? Multiplication is not allowed in this
table.

1 × 8 = 8
11 × 88 = 968
111 × 888 = 98568
1111 × 8888 = 9874568
11111 × 88888 = ...................

111111 × 888888 = ...................
1111111 × 8888888 = ...................
11111111 × 88888888 = 987654301234568
13. The following is a pattern of numbers that involves the multiplication of numbers 1, 12, 123, ….,
123456789 by 8 and adding the number 1, 2, …, 9 to it. Fill in the missing numbers to complete the
pattern.
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987

1234 × 8 + 4 = 9876
12345 × 8 + 5 = ...................
123456 × 8 + ..... = 987654
1234567 × 8 + 7 = ...................
12345678 × 8 + 8 = ...................

123456789 × 8 + 9 = 987654321

17 Patterns in Mathematics