Question

this triangular arrangement, known as pascals triangle, generates numbers base numbers.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 1
a. 2, 3, 4, 5, 6, 7
b. 8, 21, 35, 21, 20, 8
c. 7, 20, 35, 35, 20, 7
d. 7, 21, 35, 35, 21, 7

Explanation:

Step1: Recall Pascal's Triangle Rule

In Pascal's triangle, each number (except the 1s on the edges) is the sum of the two numbers directly above it. The \(n\)-th row (starting from row 0) corresponds to the coefficients of \((a + b)^n\), and the first and last elements of row \(n\) are 1, with the \(k\)-th element (0 - based) in row \(n\) given by \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\). We need to find the 8th row (since the first row is row 0: [1], row 1: [1,1],..., row 7: [1, 6, 15, 20, 15, 6, 1], so the next row is row 8).

Step2: Calculate Elements of Row 8

• First element: \(1\) (as always).
• Second element: Sum of the first two elements of row 7: \(1+6 = 7\).
• Third element: Sum of the second and third elements of row 7: \(6 + 15=21\).
• Fourth element: Sum of the third and fourth elements of row 7: \(15 + 20 = 35\).
• Fifth element: Sum of the fourth and fifth elements of row 7: \(20+15 = 35\).
• Sixth element: Sum of the fifth and sixth elements of row 7: \(15 + 6=21\).
• Seventh element: Sum of the sixth and seventh elements of row 7: \(6 + 1 = 7\).
• Eighth element: \(1\) (as always).

So the row is \([1, 7, 21, 35, 35, 21, 7, 1]\), and the missing elements (between the two 1s) are \(7, 21, 35, 35, 21, 7\).

Answer:

d. 7, 21, 35, 35, 21, 7