Solving the Polynomial Sequence: Finding First Two Terms of n^2+1

Q: Do I always start with n = 1 for sequences?

Usually yes! Unless the problem specifies otherwise, start with n = 1 for the first term, then n = 2 for the second term, and so on. This is the standard convention.

Q: What if I get confused about which term is which?

Make a simple table! Write n = 1, 2, 3... in one column and calculate each result in another column. This keeps everything organized and clear.

Q: How do I know I calculated correctly?

Double-check your arithmetic step by step. For $ n^2+1 $ with n = 2: first calculate 2² = 4, then add 1 to get 5.

Q: Can sequences have negative terms?

Absolutely! Even though this particular sequence $ n^2+1 $ gives only positive results, many sequences can have negative, zero, or fractional terms.

Q: What's the difference between a sequence and a series?

Great question! A sequence is a list of terms (like 2, 5, 10, 17...), while a series is the sum of those terms. This problem is actually about a sequence, even though it says 'series'.

Polynomial Sequences with Substitution Method

For the series $n^2+1$

Find the first two terms.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Alright, let's find the first two elements in the sequence.

00:09 We'll start by substituting the position we want into the sequence formula. Let's solve it step by step.

00:15 Remember, always calculate exponents first to get the correct result.

00:21 Great job! That's the first element in our sequence.

00:25 Now, let's use the same steps to find the second element.

00:29 Again, substitute the position into the formula, and solve it carefully.

00:36 Make sure you tackle the exponents first here as well.

00:41 And there you have it! That's how we solve this question. Nice work!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

For the series $n^2+1$

Find the first two terms.

Step-by-step solution

To solve this problem, we determine the terms of the series using the given formula $n^2 + 1$ .

We'll evaluate this series for the first two positive integer values of $n$ .

For $n = 1$ , the term is $1^2 + 1 = 1 + 1 = 2$ .
For $n = 2$ , the term is $2^2 + 1 = 4 + 1 = 5$ .

Hence, the first two terms of the series are 2 and 5. Among the choices provided, the correct answer is 2,5.

Final Answer

2,5

Key Points to Remember

Essential concepts to master this topic

Formula: Substitute consecutive positive integers starting with n = 1
Technique: For $n^2+1$ , calculate $1^2+1=2$ and $2^2+1=5$
Check: Verify pattern: first term = 2, second term = 5 ✓

Common Mistakes

Avoid these frequent errors

Starting with n = 0 instead of n = 1
Don't substitute n = 0 first to get 0²+1 = 1 as the first term! Most sequence problems expect the first positive integer values. Always start with n = 1 unless specifically told otherwise.

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

Do I always start with n = 1 for sequences?

Usually yes! Unless the problem specifies otherwise, start with n = 1 for the first term, then n = 2 for the second term, and so on. This is the standard convention.

What if I get confused about which term is which?

Make a simple table! Write n = 1, 2, 3... in one column and calculate each result in another column. This keeps everything organized and clear.

How do I know I calculated correctly?

Double-check your arithmetic step by step. For $n^2+1$ with n = 2: first calculate 2² = 4, then add 1 to get 5.

Can sequences have negative terms?

Absolutely! Even though this particular sequence $n^2+1$ gives only positive results, many sequences can have negative, zero, or fractional terms.

What's the difference between a sequence and a series?

Great question! A sequence is a list of terms (like 2, 5, 10, 17...), while a series is the sum of those terms. This problem is actually about a sequence, even though it says 'series'.

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