Answer:
1,150 pages
Step-by-step explanation:
To determine the total number of pages read in a 20-day reading challenge, where the number of pages read increases by 5 pages each day starting from 10 pages on the first day, we can use sigma notation.
First, create a formula for the number of pages read on each day, aₙ, by using the general form of an arithmetic sequence:
[tex]\boxed{\begin{array}{l}\underline{\textsf{General form of the $n$th term of an arithmetic sequence}}\\\\a_n=a+(n-1)d\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a_n$ is the nth term.}\\ \phantom{ww}\bullet\;\textsf{$a$ is the first term.}\\\phantom{ww}\bullet\;\textsf{$d$ is the common difference between terms.}\\\phantom{ww}\bullet\;\textsf{$n$ is the position of the term.}\\\end{array}}[/tex]
In this case:
So, the number of pages read on the nth day (aₙ) is:
[tex]a_n=10+(n-1)5\\\\a_n=10+5n-5\\\\a_n=5n+5[/tex]
where n represents the day number.
To find the total number of pages read over 20 days, we sum up aₙ from n = 1 to n = 20 using sigma notation:
[tex]\displaystyle \sum_{n=1}^{20} 5n+5[/tex]
Simplify this expression:
[tex]\displaystyle \sum_{n=1}^{20} 5n+\sum_{n=1}^{20}5\\\\\\5\sum_{n=1}^{20} n+\sum_{n=1}^{20}5[/tex]
Now, we can apply the formulas for the sum of the first n natural numbers and for the sum of a constant:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Sum of the first $n$ natural numbers}}\\\\\displaystyle \sum^n_{i=1} i=\dfrac{n(n+1)}{2}\end{array}}[/tex] [tex]\boxed{\begin{array}{c}\underline{\textsf{Sum of a constant}}\\\\\displaystyle \sum^n_{i=1} c=nc\end{array}}[/tex]
Therefore:
[tex]\displaystyle 5 \sum_{n=1}^{20} n + \sum_{n=1}^{20} 5 \\\\\\\\ 5 \left( \frac{20(20 + 1)}{2} \right) + 5(20)\\\\\\5 \times 210 + 100\\\\\\ 1050 + 100 \\\\\\ 1150[/tex]
So, the total number of pages read in the 20-day reading challenge is:
[tex]\Large\boxed{\boxed{\sf 1150 \;pages}}[/tex]
