Respuesta :
Sum of an arithmetic sequence is defined by the formula,
[tex]T_n=\frac{n}{2}[2a+(n-1)d][/tex]
Total length of the wooden boards will be [tex]14\frac{1}{6}[/tex] feet.
It's given in the question,
- Width of the bottom stair = [tex]3\frac{1}{2}[/tex] feet
- Width of the top stair = [tex]2\frac{1}{6}[/tex] feet
- Each stair is 4 inch shorter than the one below.
Since, 1 inch = [tex]\frac{1}{12}[/tex] feet
Therefore, [tex]4[/tex] inch = [tex]\frac{4}{12}[/tex] feet
= [tex]\frac{1}{3}[/tex] feet
So, each stair is [tex]\frac{1}{3}[/tex] feet shorter than one below.
Now the sequence formed will be,
[tex]\frac{7}{2},[\frac{7}{2}-\frac{1}{3}],[\frac{7}{2}-\frac{2}{3}],......\frac{13}{6}[/tex]
[tex]\frac{21}{6},\frac{19}{6},\frac{17}{6}....\frac{13}{6}[/tex]
It's an Arithmetic sequence with common difference 'd' = [tex]-\frac{1}{3}[/tex] and first term 'a' = [tex]\frac{7}{2}[/tex] and nth term [tex]T_n=\frac{13}{6}[/tex]
If there are 'n' stairs, expression for the for the top stair of the arithmetic sequence will be,
[tex]T_n=a+(n-1)d[/tex]
[tex]\frac{13}{6}=\frac{21}{6}+(n-1)(-\frac{1}{3})[/tex]
[tex]\frac{13}{6}-\frac{21}{6}=-(n-1)\frac{1}{3}[/tex]
[tex]\frac{8}{6}=(n-1)\frac{1}{3}[/tex]
[tex]\frac{4}{3}=(n-1)\frac{1}{3}[/tex]
[tex](n-1)=4[/tex]
[tex]n=5[/tex]
Therefore, there are 5 stairs.
Now total length of the wooden boards = Sum of 5 terms of the sequence
= [tex]\frac{n}{2}[2a+(n-1)(d)][/tex]
= [tex]\frac{5}{2}[2(\frac{21}{6})+(5-1)(-\frac{1}{3})][/tex]
= [tex]\frac{5}{2}[\frac{21}{3}-\frac{4}{3}][/tex]
= [tex]\frac{17}{3}\times \frac{5}{2}[/tex]
= [tex]\frac{85}{6}[/tex]
= [tex]14\frac{1}{6}[/tex] ft
Therefore, total length of the wooden boards will be [tex]14\frac{1}{6}[/tex] ft.
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