Question

The sum of the elements in the sixth row of pascal triangle is (1) 32 (2) 63 (3) 128 (4) 64

Short answer

Answer: 32

Step-by-step solution

Calculate the row number

To find the sum of the elements in the sixth row of Pascal's triangle, we first need to determine the row number. Since rows are numbered starting from 0, the row number for the sixth row is 5 (0, 1, 2, 3, 4, 5).

Use the power of 2 formula

To find the sum of the elements in the nth row of Pascal's triangle, we can use the formula 2^n, where n is the row number. In this case, we determined the row number to be 5.

Calculate the sum

Now that we know the row number, we can calculate the sum of the elements in the sixth row of Pascal's triangle. Using the formula 2^n, we have 2^5 = 32. The sum of the elements in the sixth row of Pascal's triangle is 32, which matches option (1) 32.

Key concepts

Sum of Elements in Pascal's Triangle

Pascal's Triangle is an intriguing mathematical concept that displays properties and patterns which have direct applications in combinatorics and probability. Each row of Pascal's Triangle represents the coefficients in the expansion of a binomial expression raised to the corresponding power. But an interesting feature of Pascal's Triangle is that the sum of the elements in any given row is a power of 2.

Specifically, if you're examining row n (remember, we start counting from row 0), the sum of all the elements in that row will be equal to \(2^n\). Therefore, when looking at the 6th row, which is row 5 in our zero-based counting system, the sum is \(2^5 = 32\). This pattern provides a quick way to calculate the sum without having to add up each individual element.

Power of 2 Formula

The power of 2 formula is a fundamental principle in mathematics, representing exponential growth. When applied to Pascal's Triangle, it asserts that the sum of the elements in the nth row equals \(2^n\). This relationship stems from the fact that each row in Pascal's Triangle corresponds to the coefficients in the binomial expansion of \((1+1)^n\), which simplifies to \(2^n\).

For instance, to find the sum of elements in any row of Pascal's Triangle quickly, you can simply take the row number (again starting from 0) and use it as the exponent with base 2. This exponential formula is not only an exercise in pattern recognition but is also a practical shortcut; you bypass the need to add long strings of numbers, instead of using a simple multiplication.

Mathematical Induction

Understanding Mathematical Induction

Mathematical induction is a proof technique used to establish the validity of statements for all natural numbers. This method involves two critical steps: the base case and the inductive step. In the base case, you prove that the statement holds true for the first natural number (usually 1 or 0). In the inductive step, you assume the statement holds for some number \(k\), then prove that it must therefore hold for \(k+1\).

Applying Mathematical Induction

We can apply this technique to prove our earlier assertion about the sum of elements in Pascal's Triangle. First, we show the base case: for the 0th row, the sum is \(2^0 = 1\), which is true. Then, for the inductive step, if we assume the sum for the nth row is \(2^n\), adding up all entries, we must show the sum for the \(n+1\)th row is \(2^n \times 2 = 2^{n+1}\). This process enforces the concept and helps learners grasp the truth behind it beyond just recognizing the pattern.