Respuesta :
To find how many terms of this sequence must be added to get 1440, we must know the arithmetic sequence equation to find any term in the sequence:
⇒ [tex]a_{n}=a_{1} +(n-1)d[/tex]
- [tex]a_{n}[/tex] --> value of the nth number of the sequence
- [tex]a_{1}[/tex] --> first term of the sequence
- n --> position of the nth term
- d --> common difference
Let's examine the information given:
- first term a = 5
⇒ [tex]a_{1} =5[/tex]
- common difference d = 2
⇒ d = 2
Therefore the equation for finding the nth term of the sequence so far is:
[tex]a_{n}=5+(n-1)*2=5+2n-2=3+2n[/tex]
Now we want to find how many terms this sequence must be added to get 1440
General equation for adding all the terms = [tex]\frac{n(a_{1}+a_{n} ) }{2}[/tex]
- [tex]a_{1}[/tex]: first term of the sequence
- [tex]a_{n}[/tex]: last term of the sequence
- n: number of terms in the sequence.
Using all the information given, let's plug in the all the values:
[tex]1440=\frac{n*(5+(3+2n))}{2} \\1440=\frac{n*(8+2n)}{2} \\2880=8n+2n^2\\2n^2+8n-2880=0\\(2n+80)(n-36)=0[/tex]
To solve this we set (2n + 80) = 0 and (n-36) = 0
[tex]2n+80=0\\ 2n = -80\\n= -40[/tex] [tex]n - 36 =0\\n = 36[/tex]
In this case, n can only equal 36 since n cannot be negative.
Answer: 36 terms of this sequence must be added to get 1440.
Hope that helps!
General formula
- a_n=a+(n-1)d
- s_n=n/2[2a+(n-1)d
We need n
- n/2[2(5)+2(n-1)]=1440
- n[10+2n-2]=2880
- n[2n+8]=2880
- 2n²+8n-2880=0
- n²+4n-1440=0
- (n+40)(n-36)=0
Take it positive
- n=36
