Problem 16 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\left\\{n^{2}+1\right\\}\)

Step-by-Step Solution

Answer

The first five terms of the sequence are 2, 5, 10, 17, and 26.

Step 1: Identify the Sequence Formula

The given sequence is defined by the formula \{s_{n}\right\}=\(n^2 + 1\). This formula will be used to calculate each term of the sequence for different values of \(n\).

Step 2: Calculate the First Term (=1\right\)

Plug in \(n = 1\) into the sequence formula: \(s_1 = 1^2 + 1 = 1 + 1 = 2\). So, the first term is 2.

Step 3: Calculate the Second Term (=2\right\)

Plug in \(n = 2\) into the formula: \(s_2 = 2^2 + 1 = 4 + 1 = 5\). Thus, the second term is 5.

Step 4: Calculate the Third Term (=3\right\)

Plug in \(n = 3\) into the formula: \(s_3 = 3^2 + 1 = 9 + 1 = 10\). Hence, the third term is 10.

Step 5: Calculate the Fourth Term (=4\right\)

Plug in \(n = 4\) into the formula: \(s_4 = 4^2 + 1 = 16 + 1 = 17\). Therefore, the fourth term is 17.

Step 6: Calculate the Fifth Term (=5\right\)

Plug in \(n = 5\) into the formula: \(s_5 = 5^2 + 1 = 25 + 1 = 26\). So, the fifth term is 26.

Key Concepts

sequence formula

In the world of algebra, a sequence is essentially an ordered list of numbers. These numbers follow a particular pattern, which is usually defined by a formula. This formula is often expressed in terms of the position of numbers in the sequence, denoted by 'n'.
The sequence formula tells us how to calculate each term in the sequence. In our given exercise, the sequence formula is \(s_n = n^2 + 1\). This means that to find any term in this sequence, you simply take the position of that term (n), square it, and then add 1.
Understanding the sequence formula is crucial because it unlocks the ability to find any term in the sequence without listing all the preceding terms. It provides a straightforward way to generate terms directly.

term calculation

Term calculation involves finding specific terms in the sequence using the sequence formula. In our exercise, we need to calculate the first five terms using the formula \(s_n = n^2 + 1\). Let’s break it down step-by-step:

• First term: Substitute n = 1 into the formula: \(s_1 = 1^2 + 1 = 2\). The first term is 2.
• Second term: Substitute n = 2 into the formula: \(s_2 = 2^2 + 1 = 5\). The second term is 5.
• Third term: Substitute n = 3 into the formula: \(s_3 = 3^2 + 1 = 10\). The third term is 10.
• Fourth term: Substitute n = 4 into the formula: \(s_4 = 4^2 + 1 = 17\). The fourth term is 17.
• Fifth term: Substitute n = 5 into the formula: \(s_5 = 5^2 + 1 = 26\). The fifth term is 26.

Term calculation is often about simply applying the sequence formula repeatedly for different values of n.

nth term in sequence

The nth term in a sequence refers to the general formula that defines any term's position in the sequence. In our exercise, the nth term is expressed as \(s_n = n^2 + 1\).
To determine the nth term in a sequence, follow these steps:

• Identify the pattern or rule of the sequence. In many algebraic sequences, this will be given as a formula.
• Substitute the position number (n) into the sequence formula. This directly gives you the term at that position.
• For instance, if you wanted to find the 10th term in our sequence, you would substitute n = 10 into the formula: \(s_{10} = 10^2 + 1 = 101\).

Recognizing the nth term formula enables you to predict the value of any term in the sequence quickly and without the need to manually calculate all preceding terms.