Question

1. find \\( \sum\limits_{x=1}^{8} 2x + 3 \\)

Explanation:

Step1: Recall the summation formula

The sum \(\sum_{i = 1}^{n} (a_i + b_i)=\sum_{i = 1}^{n}a_i+\sum_{i = 1}^{n}b_i\), and \(\sum_{i = 1}^{n}ca_i = c\sum_{i = 1}^{n}a_i\) for constant \(c\), also \(\sum_{i = 1}^{n}c=cn\) where \(c\) is a constant. Here we have \(\sum_{x = 1}^{8}(2x + 3)\), so we can split it into \(\sum_{x = 1}^{8}2x+\sum_{x = 1}^{8}3\).

Step2: Calculate \(\sum_{x = 1}^{8}2x\)

Using the formula \(\sum_{i = 1}^{n}ca_i=c\sum_{i = 1}^{n}a_i\), we get \(2\sum_{x = 1}^{8}x\). The formula for \(\sum_{i = 1}^{n}i=\frac{n(n + 1)}{2}\), with \(n = 8\), so \(\sum_{x = 1}^{8}x=\frac{8\times(8 + 1)}{2}=\frac{8\times9}{2}=36\). Then \(2\sum_{x = 1}^{8}x=2\times36 = 72\).

Step3: Calculate \(\sum_{x = 1}^{8}3\)

Using the formula \(\sum_{i = 1}^{n}c=cn\) with \(c = 3\) and \(n = 8\), we get \(3\times8=24\).

Step4: Add the two results

Now we add the results from Step2 and Step3: \(72+24 = 96\).

Answer:

\(96\)