√5/4,√3/2,√7/4. Find the formula for this sequence. √ this symbol means square root

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

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.

uwais17 | Student

The question states that we need to determine the formula of the sequence. There are generally three types of sequences:

 

  1. Arithmetic: Common Difference 
  2. Geometric: Common Ratio
  3. 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)

where

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)

 

 

 

 

 

 

 

 

 

ogutu | Student

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.