# Finding the formula for a sequence of terms (2023)

## Introduction

👉 Learn how to write the rule of a sequence given a sequence of numbers. A sequence is a list of numbers/values exhibiting a defined pattern. A number/value in a sequence is called a term of the sequence.

To write the explicit formula of a sequence of numbers, we first determine whether each term of the sequence is obtained by adding/subtracting a common value to the preceding term or whether each term of the sequence is obtained by multiplying a pre-determined value, called the common ratio, to the preceding term or neither.

For the first case, the sequence is an arithmetic sequence and we can write the explicit formula by plugging the common difference into the arithmetic sequence explicit formula. For the second case, the sequence is a geometric sequence and we can write the explicit formula by plugging the common ratio into the geometric sequence explicit formula. For the third case, we can examine the pattern and by some kind of trial and error, we can be able to write the explicit formula.

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✅ Sequences
✅Determine The First Five Terms of The Arithmetic Sequence
✅How to Write The Formula for a Arithmetic Sequence
✅Find the nth Term of an Arithmetic Sequence
✅Find the First Five Terms of a Geometric Sequence
✅How to Write The Formula for a Geometric Sequence
✅Find the nth Term of a Geometric Sequence
✅How to Determine Arithmetic or Geometric Sequence
✅Find the First Five Terms of a Sequence
✅How to Write The Formula for a Sequence
✅Find the nth Term of a Sequence
✅How to Simplify Factorials
✅Recursive Sequences
✅Prove the Sum by Induction
✅Find the Given Term of Binomial Expansion
✅How to Expand a Binomial

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## Content

And what you get, yes, yes, exactly all right.

So now we got an issue.

One thing we noticed, ladies and gentlemen, is we go from positive to negative right now when we looked at those formulas, what did we have when we went from a positive to a negative? How did we do that? Why don't we we had an exponent right? But what we add? What raised to a power? We had a negative one.

So we know that there's probably going to be a negative 1 raised to it and power right? We know that is going to happen.

Now what we want to do is we want to look at this, and we want to say all right so what's happening to my values.

Okay, how am I going from one term to the next? When I plug in my terms and I'm getting my ending value? What is the relationship? And the first thing I was like to look at look at arithmetic relationships.

What am I adding or subtracting? So you could say, okay and a 2 2, a 2 negative 5, right? I'm, subtracting 7, well, do I subtract 7 again and get to 10, no, no I'm.

Well.

Now, I'm, adding 15 to go from negative 5 to 10 so we're, not adding or subtracting the same amount of numbers to go from one to the next right? Then we look at multiplication.

And we see that you know when I plug in one of these numbers, if I plug in 2 can I multiply, this 2 by something to be able to get me to the negative 5, and then add or subtract something well, I could say, huh, you're gonna have to multiply by a negative number.

But you also have this negative 1 raised to the N.

So what so that's, what I'm saying, if you did too right, if you multiplied it by negative, two right and then subtracted it from one right that'll, give you negative 5 2 times negative 2, minus 1.

Negative 2 is negative 4.

And then minus 1 is 5.

All right now.

Let's, see if that would work for this 2 times 3 is 6, minus 1 is 7.

Well, that's, not even getting me close right.

I'm.

Not worried about the alternate signs because that's just gonna alternate signs, but now I'm, not even close to that right.

And then 4 times 2, that's, not gonna work.

So now I need to think of an if different relationship, how can I get from these values.

So we kind of marked off adding and subtracting right? I, can't, add or subtract.

My terms I can't when I plug in 4 I, don't know, why I was looking at these terms, rigid, I, can't, add or subtract 3 to get 10 because it's, not the same so there's.

Another relationship, adding and subtracting are out multiplication and dividing are out what's another way that we could what's another relationship or another operation.

Well, adding and subtraction are out.

We can't, add or subtract the same number for each one of these terms, well, it could take square root.

But these know, if I take the square root of 4 that's 2, that's, not close to 17, but what's, the opposite of square root, opposite of square roof, squaring.

So let's, see if squaring does that get us anywhere close if I took three and squared it, that's, nine.

What I have to do to get to ten and one right would that give me close here if I did four times, four squared would be 16, plus one would be 17.

It is negative.

But remember we have the negative component here right? What are we getting? I know? But I'm just saying are we getting close to our numbers to two squared? Plus one is five right so that that's something good.

So we have N squared plus one.

Now we just need to make sure how is this going to work all right? So let's see does this formula? Work let's do a sub 1.

A sub 1 equals negative 1 to the first power times 1 squared, plus 1.

Well, negative 1 squared is negative 1 1 squared is 1, plus 1 is 2.

So that equals a negative 2, we need a 2 right? Not a negative 2.

So I'm close, but I have the wrong sign.

So something I need to do to maybe change this.

So maybe what I could do is.

So since this is starting off a negative that means I'm starting as an even as a odd power.

So I probably want to start this as a even-par.

So what could I rewrite? Instead of n, I could write it as n? How can I make an odd - and even I just have to add how many values 1 so let's try this and plus 1.

So now let's do a sub 1 negative 1 times 1, plus 1 times 1 squared, plus 1 1, plus 1 is 2.

Negative 1 squared is or negative 1 squared is 1 times 1 squared is 1.

Plus 1 is 2 looks pretty good right? Let's.

Try another one just make sure it works.

Correct let's do a sub 2.

So if I do a c2 I have negative 1 2, plus 1 times 2 squared, plus 1 2, plus 1 is 3.

Negative 1 cubed is negative 1 2 squared is 4.

Plus 1 is 5.

Are we think like we're on the right path? Ok, yes.

Yes.

We're.

Not really guessing, we're, just kind of working through what the relationship is I'm going through there and yes, that's.

What that's exactly what I was trying to describe to you guys is ladies and gentlemen.

The first thing we learned.

## FAQs

### How do you find the formula for the term of a sequence? ›

Solution: To find a specific term of an arithmetic sequence, we use the formula for finding the nth term. Step 1: The nth term of an arithmetic sequence is given by an = a + (n – 1)d.

What is the sum of the expression 1 2 3 4 5 6 7 up to 100 terms? ›

According to arithmetic progression, natural numbers can be written down as 1, 2, 3, 4, 5, 6, 7, and 8 to 100. Basically, the sum of the first 100 natural numbers is equal to 5050.

How to find the formula of a sequence without common difference? ›

How do you find the nth term formula for a sequence with a non-constant difference?
1. Calculate the difference between the terms. 8 - (-3) = 11. ...
2. Calculate 2n2: ...
3. Subtract the answers of 2n2 from original sequence: ...
4. Calculate the difference of the sequence -5, 0, 5, 10, 15: ...
5. Calculate a linear sequence using an = a + (n -1) d.

How to make 100 with 1 2 3 4 5 6 7 8 9? ›

123 + 4 - 5 + 67 - 89 = 100.

Here are the rules: use every digit in order - 123456789 - and insert as many addition and subtraction signs as you need so that the total is 100.

What is the series 1 2 3 4 5 6 to 100? ›

The natural numbers from 1 to 100 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, ...

What is the formula for 1 2 3 4 5? ›

This is an arithmetic sequence since there is a common difference between each term. In this case, adding 1 to the previous term in the sequence gives the next term. In other words, an=a1+d(n−1) a n = a 1 + d ( n - 1 ) . This is the formula of an arithmetic sequence.

How do you find the missing term number sequence? ›

Step 1: Find the common difference of each pair of consecutive terms in the sequence by subtracting each term from the term that comes directly after it. Step 2: Add the common difference to the number prior to the first missing number in the sequence. Step 3: Repeat Step 2 for any other missing numbers.

What is the term to term rule? ›

A term by term rule is used for a sequence in which the next term is obtained from the previous term. Example: Arithmetic sequence. In an arithmetic sequence, each term (other than the first term) is obtained by adding or subtracting a constant value from the preceding term.

What is the formula for sequence and series? ›

Arithmetic Sequence and Series Formulas

nth term of arithmetic sequence, an = a + (n - 1) d. Sum of the arithmetic series, Sn = n/2 (2a + (n - 1) d) (or) Sn = n/2 (a + an)

What is the 4 types of sequence? ›

There are four main types of different sequences you need to know, they are arithmetic sequences, geometric sequences, quadratic sequences and special sequences.

### What is an example of a term to term rule? ›

For example, the odd numbers (1,3,5,7,9…) are a sequence. Each individual number in a sequence is a term in the sequence, and we'll need to find the operation, such as adding or subtracting a number that will allow us to work out the next number in the sequence. This rule is known as the term-to-term rule.

What is the math pattern 1 1 2 3 5 8? ›

What is the Fibonacci sequence? The Fibonacci sequence is a famous group of numbers beginning with 0 and 1 in which each number is the sum of the two before it. It begins 0, 1, 1, 2, 3, 5, 8, 13, 21 and continues infinitely.

What is the number 1 2 3 4 5 6 7 8 9? ›

Answer: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.

How many three digit numbers can generated from 1 2 3 4 5 6 7 8 9 such that the digits are in ascending order? ›

Thus a total of: 28 + 21 + 15 + 10 + 6 + 3 + 1 = 84 such numbers.

What is the 100th term of the sequence 1 2 2 3 3 3 4 4 4 4? ›

Hence, 100th term is 14.

What is the sum of series 1 1 3 1 5 1 7? ›

The sum of series 1 + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 is 2.133256. Explanation: The sum of series 1 + 1/3 + 1/5 + 1/7 + … + 1/41 is 2.479674.

What is the sum of the following series 1 3 5 7 upto 12 terms? ›

upto 12 terms. The arithmetic progression 1, 3, 5, 7, …. is given. Here first term is 1 and common difference is 3 – 1 = 2. Hence, Sum of first 12 terms is 144.

What is the name of 1 2 3 4 5 6 7 8 9 10? ›

Natural Numbers The numbers that we use when we are counting or ordering {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 …} Whole Numbers The numbers that include natural numbers and zero. Not a fraction or decimal.

What is the value of 6 if 1 3 2 3 3 5 4 4 and 5 4? ›

​If 1=3,2=3,3=5,4=5,5=4, So , 6=4 is the right answer, because in series 3,3,5,5,4,4 so 1=3, 2=3, 3=5, 4=5, 5=4 and 6=4.

What is the value of 6 if 1 3 2 3 3 5 4 5 and 5 4? ›

We can observe here that the puzzle is on the logic of number written in words. So, the number 6 will be equal to three as it has three letters in its word. Hence, The required answer is 3.

### What is sequence in math grade 4? ›

A sequence is a list of numbers in a certain order. Each number in a sequence is called a term . Each term in a sequence has a position (first, second, third and so on).

What is the missing term in the sequence 1 4 9 16 36 49? ›

Detailed Solution

Hence, "option 4" is the correct answer.

What is the next number in the series 2 1 1 2 1 4? ›

Hence, the correct answer is "1/8".

What is the next number in the sequence 1 2 4 7 ___ ___ 22? ›

Answer: The number that fits best in the sequence 1, 2, 4, 7, 11, …, 22 is 16.

What is the formula for the number pattern? ›

A linear number pattern is a list of numbers in which the difference between each number in the list is the same. The formula for the nth term of a linear number pattern, denoted an, is an = dn - c, where d is the common difference in the linear pattern and c is a constant number.

What is the next number in sequence 1 1 2 4 3 9 4? ›

Solution: Given, sequence 1, 1, 2, 4, 3, 9, 4, …… Hence, the next number is 55.

What is the sum of the numbers 1 to 100? ›

∴ Sum of first 100 natural numbers is 5050.

What is the sum of series 1 2 3 3 5 4 7 upto 11 terms? ›

or, S=946.

What is the sum to 100 terms of 1 2 3 4? ›

Sum of 1+2+3+4...... 100 is 5050. Hope it will help you !!!

What is the sum of 1st 100 terms 1 4 6 5 11 6? ›

Hence the sum of first 100 terms of the given sequence is 7600.

### What is one plus two plus three plus four etc? ›

For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.

What is the 20th even number after 284? ›

Even Numbers from 1 to 1000
2420
242244260
282284300
322324340
362364380
18 more rows

What is the total if you add 1 to 200? ›

Hence, the answer to this question is 20100. Note – In this problem you need to consider the series from 1 to 200 as an AP and use the formula of sum of 200 terms of an AP.

What is the sum of the following series 1 2 3 4 5 100? ›

+ 100 = 5050 .

What is the sum of arithmetic progression 2 5 8 11 to 12 terms? ›

Hence, the sum of the given terms is 610.

What is the sum of first 40 terms of 1 3 4 5? ›

Detailed Solution

The series can be divided into two, sum of odd place terms and even place terms for easy addition. ∴ The sum of the first 40 terms is 1030.

What will be the sum of 1 2 3 4 5 6? ›

Answer: The sum of the series 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 is equal to 45.

What is the sum of 200 terms of the series 1 4 6 5 7 6? ›

Hence, the sum of 200 terms to the given series is 30,200.

What is the sum of first 30 terms of the sequence 1 2 4 7 11? ›

Statement - 1: The sum of the first 30 terms of the sequence 1,2,4,7,11,16,22,29,37,46...is 4520 Statement - 2: The successive differences of the terms of the sequence form an AP O Mark for review later Statement - 1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement -1 Statement -1 is ...

What is the sum of numbers from 1 to 1000? ›

Thus, the sum of the first 1000 positive integers is 500500.

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