<h3>
Answer: 17</h3>
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Work Shown:
a1 = 3 = first term
d = 2 = common difference (since we add 2 to each term to get the next one)
Let's compute the nth term.
an = a1 + (n-1)*d
an = 3 + (n-1)*2
an = 3 + 2n-2
an = 2n+1
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To check things so far, we can plug in something like n = 2
an = 2n+1
a2 = 2*2+1
a2 = 5
Showing that the 2nd term is 5, which matches with the sequence given to us
Let's check n = 3
an = 2n+1
a3 = 2*3+1
a3 = 7
That matches as well. I'll let you check the others.
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Plug in n = 8 to find the 8th term
an = 2n+1
a8 = 2*8+1
a8 = 17
The eighth term is 17, which is the final answer.
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You could extend out the given sequence by adding 2 each time until you reach the 8th term
3,5,7,9,11,13,15,17
Though this method is slow if you need to find say the 38th term
In mathematics, number sequencing of the same pattern are called progression. There are three types of progression: arithmetic, harmonic and geometric. The pattern in arithmetic is called common difference, while the pattern in geometric is called common ratio. Harmonic progression is just the reciprocal of the arithmetic sequence.
The common ratio is denoted as r. For values of r<1, the sum of the infinite series is equal to
S∞ = A₁/(1-r), where A1 is the first term of the sequence. Substituting A₁=65 and r=1/6:
S∞ = A₁/(1-r) = 65/(1-1/6)
S∞ = 78
The given inequality is:
This inequality can be divided in two parts as:
a)
b)
Solving part a:
Solving part b:
Therefore, the solution to the given inequality is
and
. Combining both the ranges we get the solution:
.
In interval notation, this solution can be expressed as [1,5]
