√5/4,√3/2,√7/4. Find the formula for this sequence. √ this symbol means square root
I think that the root sign acts on entire fractions, i.e. we have the sequence
`sqrt(5/4),` `sqrt(3/2),` `sqrt(7/4).`
Let's express the second fraction as `6/4` and the sequence becomes
`sqrt(5/4), sqrt(6/4), sqrt(7/4).`
Now the rule is obvious: n-th term is `sqrt((n+4)/4)` if we start from `n=1.` This is the same as `sqrt(1+n/4).`
That said, there are infinitely many possible formulas for these three numbers, even among polynomial formulas.
The question states that we need to determine the formula of the sequence. There are generally three types of sequences:
- Arithmetic: Common Difference
- Geometric: Common Ratio
- Quadratic: Second Difference
We need to determine the type of sequence before we can determine the formula of the aforementioned sequence.
The sequence was given as: √(5/4), √(3/2), √(7/4).
We need to change the sequence into decimal form as it is difficult to find the pattern in the fraction form.
The sequence in decimal form: √1.25, √1.5, √1.75
If we ignore the root we have: 1.25, 1.5, 1.75
From above we can see a clear between and that there is a common difference of 0.25
Since there is a common difference we have identified the sequence to be arithmetic.
The formula for an arithmetic sequence is as follows:
Tn = a + d*(n-1)
Tn: Term value
a: first term
n: Term number
So the formula of the sequence is as follows
The first term is a = 1.25, d = 0.25 and do not forget the foot:
Tn = √[1.25 + 0.25 (n-1)]
Now we know the sequence of our pattern, let double check our formula
T1=√[1.25 + 0.25 (1-1)]= √1.25 = √(5/4)
T2 = √[1.25 +0.25(2-1)] = √1.5 = √(3/2)
T3 = √[1.25 +0.25 (3-1)] = √1.75= √(7/4)
We have been given three numbers in a sequence. They are √(5/4) √(3/2) and √(7/4).
It is hard to see any connection between the numbers while they are in fraction form.Therefore, we will change the numbers from fraction form to decimal form.
√(5/4) becomes √1.25
√(3/2) becomes √1.5
√(7/4) becomes √1.75
If we look at the numbers closely, we see that the difference from one to the other is √0.25. Let us confirm this.
√1.25 - √1.5 =- √0.25
√1.5- √1.75= -√0.25
This means that in order to move to the right of the sequence we have to subtract √0.25 from the preceding number.