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

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

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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