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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

58. DISCOVER PROVE: Sums of Binomial Coefficients Add each of the first five rows of Pascal’s Triangle, as indicated. Do you see a pattern?

1+1=?

1+2+1=?

1+3+3+1=?

1+4+6+4+1=?

1+5+10+10+5+1=?

On the basis of the patterns you have found, find the sum off the nth row:

C n 0 + C n 1 + C n 2 + + C n n

Prove your result by expanding ( 1 + 1 ) n using the Binomial Theorem.

To determine

To find:

The sum of the nth row.

Explanation

Given:

1+1=?1+2+1=?1+3+3+1=?1+4+6+4+1=?1+5+10+10+5+1=?

Approach:

The binomial expansion of (1+1)n is given by

(1+1)n=Cn01n10+Cn11n111+Cn21n212++Cnn101n

Calculation:

1+1=21+2+1=41+3+3+1=81+4+6+4+1=161+5+10+10+5+1=32

Here, the pattern is 2n or (1+1)n where n is the row.

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