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Common Difference Calculator + Online Solver With Free Steps

Common Difference Calculator + Online Solver With Free Steps

Common Difference Calculator + Online Solver With Free Steps

The Common Difference Calculator is an online tool for analyzing a series of numbers that are produced by repeatedly adding a constant number.

The first term, common difference, nth term, or the sum of the first n terms can all be determined with this calculator.

What Is a Common Difference Calculator?

The Common Difference Calculator is a tool that computes the constant difference between consecutive terms in an arithmetic sequence.

The common difference in an arithmetic sequence is the difference between any of its words and the term before it. An arithmetic sequence always adds (or subtracts) the same number to go from one term to the next.

The amount that is added (or removed) at each point in an arithmetic progression is referred to as the “common difference” because, if we subtract (that is if we determine the difference of) succeeding terms, we will always arrive at this common value. The letter “d” is typically used to indicate the common difference.

Consider the following arithmetic series: 2, 4, 6, 8,…

Here, the common difference between each term is 2 as:

2nd term – 1st term = 4 – 2 = 2 

3rd term – 2nd term = 6 – 4 = 2 

4th term – 3rd term = 8 – 6 = 2

and so on.

How To Use a Common Difference Calculator?

You can use the Common Difference Calculator by following the given instructions to get the value of the difference for the given sequence or series.

Step 1

Fill in the provided input boxes with the first term of the sequence, the total number of terms, and the common difference.

Step 2

Click on the “Calculate Arithmetic Sequence” button to determine the sequence of the given difference and also the whole step-by-step solution for the Common Difference will be displayed.

How Does the Common Difference Calculator Work?

The Common Difference Calculator works by determining the common difference shared between each pair of consecutive terms from an arithmetic sequence by using Arithmetic Sequence Formula.

Arithmetic Sequence Formula helps us in the calculation of the nth term of an arithmetic progression. The arithmetic sequence is the sequence where the common difference remains constant between any two successive terms.

Arithmetic Sequence Formula

Consider a case in which you need to locate the 30th term in any of the previously described sequences except for the Fibonacci sequence, of course.

It would take a long time and be laborious to write out the first 30 terms. However, you surely observed that you don’t have to record them all. If you extend the first term by 29 common differences, that is sufficient.

The arithmetic sequence equation can be created by generalizing this assertion. Any nth term in the sequence can be represented by the given formula.

a = a1 + (n-1) . d 

where:

a — The nth term of the sequence;

d — Common difference; and

a1 — First term of the sequence.

Any common difference, whether positive, negative, or equal to zero, can be calculated using this arithmetic sequence formula. Naturally, all terms are equal in the scenario of a zero difference, eliminating the need for any calculations.

Difference Between Sequence and Series

Consider the following arithmetic sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. We could manually add up all of the terms, but that isn’t necessary.

Let’s attempt to sum up the concepts more systematically. The first and last terms will be added together, followed by the second and next-to-last, third and third-to-last, etc.

You’ll observe right away that:

3 + 21 = 24 

5 + 19 = 24 

7 + 17 = 24 

Each pair’s sum is constant and equals 24. So, we don’t have to add all the numbers. Simply add the first and last terms in the series, and then divide the result by the number of pairs, or $ \frac{n}{2} $.

Mathematically, this is written as:

\[ S = \frac{n}{2} \times (a_1 + a) \]

Substituting the arithmetic sequence equation for $ n_th $ term:

\[ S = \frac{n}{2} \times [a_1 + a_1 +(n-1) \cdot d] \]

After simplification:

\[ S = \frac{n}{2} \times [2a_1 +(n-1) \cdot d] \]

This formula will allow you to find the sum of an arithmetic sequence.

Solved Examples

Let’s explore some examples to better understand the working of the 2-step calculator.

Example 1

Find the common difference between the a2 and a3, if  a1 = 23 , n = 3, d = 5?

Solution

Given a2 and a5, a1 = 23, n = 3 , d = 5 , a4 = 20 

Apply the formula,

an = a1 + (n-1)d 

a2  = 23 + (3 -1) x 5 = 23 + 10 = 33  

a5 = a4 + (n-1)d = 20 + (3-1) x 5 = 20 + 10 = 30 

d = a{n+1} – an = a2 – a5= 33 – 30 = 3 

Therefore, the common difference in an arithmetic sequence is 3.

Example 2

Determine the common difference for the arithmetic sequence given below.

  1. a) {$\dfrac{1}{3}$, $1$, $\dfrac{5}{3}$, $\dfrac{7}{3}$}
  2. b) {$\dfrac{5}{3}$,$\dfrac{8}{3}$,$\dfrac{11}{3}$,$\dfrac{14}{3}$}

Solution

a)

The given sequence is = $\dfrac{1}{3}$, $1$, $\dfrac{5}{3}$, $\dfrac{7}{3}$…

We calculate the difference between the two consecutive terms of the sequence.

\[1- \dfrac{1}{3} = \dfrac{2}{3} \]

\[\dfrac{5}{3} − 1 = \dfrac{2}{3} \]

\[\dfrac{7}{3} − \dfrac{5}{3} = \dfrac{2}{3} \]

Hence, the answer is $\dfrac{2}{3}$.

b)

The given sequence is = $\dfrac{5}{3}$,$\dfrac{8}{3}$,$\dfrac{11}{3}$,$\dfrac{14}{3}$.

We calculate the difference between the two consecutive terms of the sequence.

\[ \dfrac{8}{3} – \dfrac{5}{3} = \dfrac{3}{3} = 1 \]

\[ \dfrac{11}{3} − \dfrac{8}{3} = 1  \]

\[ \dfrac{14}{3} − \dfrac{11}{3} = 1 \]

Hence, the required answer is $1$.

Example 3

Determine the common difference of the given arithmetic sequences if the value of n = 5.

  1. a) {$6n – 6$, $n^{2}$,$ n^{2}+1$}
  2. b) {$5n + 5$, $6n + 3$, $7n + 1$}

Solution

a)

The value of n is equal to “5”, so by putting this value in the sequence we can calculate the value of each term.

6n – 6 = 6 (5) – 6 = 24 

\[ n^{2} = 5^{2} = 25  \]

\[ n^{2}+ 1 = 5^{2}+1 = 26 \]

So the sequence can be written as {24, 25, 26}.

The common difference is d= 25 – 24 = 1 or d = 26 – 25 = 1.

Alternatively, we can subtract the third term from the second.

\[ d = n^{2}+ 1 – n^{2} = 1 \].

b)

The value of n is equal to “5″, so by putting this value in the sequence we can calculate the value of each term.

5n + 5 = 5 (5) + 5 = 30

6n + 3 = 6 (5) + 3 = 33

7n + 1 = 7 (5) + 1 = 36

So the sequence can be written as {30, 33, 36}.

Then d= 33 – 30 = 3 or d = 36 – 33 = 3.

Alternatively, we can subtract the second term from the first or the third term from the second.

d = 6n + 3 – ( 5n + 5) = n – 2 = 5 – 3 = 2 

or

d = 7n + 1 – ( 6n + 3) = n – 2 = 5 – 3 = 2 

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