# Worked Examples

Numbers from 1 to 100 are arranged in the shape of a triangle as given below:

Two groups of numbers, on the left of the symbol â€˜=â€™, are given which are related in some way. You have to identify the relationship and try to establish a similar relationship between the term given on the right side of the symbol â€˜=â€™ and the missing term, which is indicated by a question mark. You have to select the correct answer among the four choices provided.
- 2 8 12 : 7 6 5 = 9 11 23 : ?
- 12 21 32
- 12 22 30
- 12 11 10
- 12 13 14

- 10 25 26 : 10 23 30 = 28 47 52 : ?
- 28 47 54
- 28 45 56
- 28 46 54
- 28 29 30

By observing the above triangle carefully, you find that number 1 is present in the first row. In the second row, you have numbers from 2 to 4, i.e., from (1^{2 }+ 1) to 2^{2}. In the third row, you have numbers from 5 to 9 written in the reverse order, i.e., from (2^{2} + 1) to 3^{2}. The fourth row has numbers from 10 to 16, i.e., from (3^{2} + 1) to 4^{2}. In this way, in the *n*th row, you have numbers from (*n* â€“ 1)^{2} + 1 to *n*^{2}. Every row either begins or ends with a number *n*^{2}. Also, alternately, the direction of numbers is from left to right and from right to left.

- The numbers in the first term of the LHS are 2, 8 and 12. They form a vertical line in the given triangle of numbers. The numbers in the second term of the LHS are 7, 6 and 5. They form a horizontal line in the triangle. Further, 8 and 7 are adjacent to each other.

These terms on the LHS form a shape as shown below:

The numbers in the first term of the RHS are 9, 11 and 13. They are in a vertical line with the number 11 in the middle.

Hence, the numbers in the second term of the RHS should be 12, 13 and 14, which are in a horizontal line after the number 11.

**Answer:** (4).
- The numbers in the first term of the LHS, 10, 25 and 26, form a diagonal line.

The numbers in the second term of LHS, 10, 23 and 30, form another diagonal, line. The first number in these terms is common. The location of these two diagonals in the number triangle is as shown below:

Similarly, the numbers in the first term of RHS, 28, 47 and 52, forms a diagonal, and this diagonal is parallel to the diagonal formed by the numbers 10, 25 and 26.

Hence, the numbers in the second term of RHS should start with 28 and must be in a line parallel to the diagonal formed by the numbers 10, 23 and 30. So, the numbers in the second term on RHS should be 28, 45 and 56.

**Answer:** (2)