Q. 84.1( 7 Votes )

# Look at the following pattern:

This is called Pascal’s triangle. What is the middle number in the 9^{th} row?

Answer :

Let us try to draw out a formula by which this Pascal’s triangle is formed.

Notice, corners always contain 1.

Also, the number appearing in the next row in the middle of two numbers from the preceding row is the sum of the numbers of the preceding row.

Following this, we have

Understand the pattern,

In the first row, the corner is occupied by 1.

In the second row, the corners are occupied by 1.

In the third row, the corners are occupied by 1, and the middle number is found by adding the two numbers on its either sides in the preceding row.

We have,

In the fourth row, the corners are occupied by 1, and the other middle numbers are found by adding the two numbers on its either sides in the preceding row.

We have,

1 (1 + 3) (3 + 3) (3 + 1) 1

In the fifth row, the corners are occupied by 1, and the middle row numbers are computed by adding the numbers on its either sides in the preceding row.

We have,

1 (1 + 4) (4 + 6) (6 + 4) (4 + 1) 1

In the sixth row, the corners are occupied by 1, and the middle row numbers are computed by adding the numbers on its either sides in the preceding row.

We have,

1 (1 + 5) (5 + 10) (10 + 10) (10 + 5) (5 + 1) 1

In the seventh row, the corners are occupied by 1, and the middle row numbers are just derived by adding the numbers on its either sides in the preceding row.

1 (1 + 6) (6 + 15) (15 + 20) (20 + 15) (15 + 6) (6 + 1) 1

In the eighth row, the corners are occupied by 1, and the middle row numbers are just derived by adding the numbers on its either sides in the preceding row.

We get

1 (1 + 7) (7 + 21) (21 + 35) (35 + 35) (35 + 21) (21 + 7) (7 + 1) 1

⇒

**Thus, the middle number in the ninth row is 70.**

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