PASCAL’S TRIANGLE For Exercises 5 and
6, use the following information.
Blaise Pascal is known for a special
triangular arrangement of numbers, calledPascal’s Triangle.. Each number insuccessive rows of the triangle is created by adding the two numbers in the row above the number.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
5. Starting with the second row from thetop, what is the sequence of the diagonal
(1 2 3 4 5 6 7)?
6. The sum of each row represents 2n,
starting with the first row, which is
20 = 1. Write an exponential expression
for the sum of the values in the sixth
row. Simplify the expression.
isn't that obvious? nth term = n
as you said, sum = 2^(n-1) for nth row. So, sum of 6th row = 2^5 = 32
To find the sequence of the diagonal (1 2 3 4 5 6 7) in Pascal's Triangle, you can use the row index and the position within that row to determine the value.
In Pascal's Triangle, each number is obtained by adding the two numbers directly above it in the row above. Considering this, we can find the value in the diagonal by taking the row index (starting with 1 as the first row) and finding the value at the same position within that row.
Let's break it down for the given diagonal sequence:
- The first number is 1, which can be found in the first row.
- The second number is 2, which can be found in the third row at position 2.
- The third number is 3, which can be found in the fourth row at position 3.
- The fourth number is 4, which can be found in the fifth row at position 4.
- The fifth number is 5, which can be found in the sixth row at position 5.
- The sixth number is 6, which can be found in the seventh row at position 6.
- The seventh number is 7, which is not present in Pascal's Triangle, as it only goes up to the sixth row.
Therefore, the sequence of the diagonal (1 2 3 4 5 6 7) can be found in Pascal's Triangle up to the sixth row.
To answer the question about the sum of values in the sixth row, we can use the formula given in the information, which states that the sum of each row represents 2^n, starting with the first row.
For the sixth row, n = 6. Therefore, the sum of values in the sixth row can be written as 2^6.
Simplifying, 2^6 = 64.
So, the exponential expression for the sum of values in the sixth row is 2^6, which simplifies to 64.