Question

math section
if the pattern below represents rows 1 - 3 of a pattern. if additional rows are added beneath the third row, what will be the total number of circles in the first 9 rows?
a 21
b 63
c 117
d 130

Explanation:

Step1: Identify the pattern

The number of circles in each row forms an arithmetic - sequence. The first row has 5 circles, the second row has 7 circles, the third row has 9 circles. The common difference \(d = 2\) and the first - term \(a_1=5\). The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). So \(a_n=5+(n - 1)\times2=2n + 3\).

Step2: Find the sum formula

The sum of the first \(n\) terms of an arithmetic sequence is \(S_n=\sum_{k = 1}^{n}a_k=\sum_{k = 1}^{n}(2k + 3)\). We know that \(\sum_{k = 1}^{n}(2k+3)=2\sum_{k = 1}^{n}k+\sum_{k = 1}^{n}3\). Since \(\sum_{k = 1}^{n}k=\frac{n(n + 1)}{2}\) and \(\sum_{k = 1}^{n}3=3n\), then \(S_n=2\times\frac{n(n + 1)}{2}+3n=n^2 + n+3n=n^2+4n\).

Step3: Calculate \(S_9\)

Substitute \(n = 9\) into the sum formula \(S_n=n^2+4n\). So \(S_9=9^2+4\times9=81 + 36=117\).

Answer:

C. 117