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Arithmetic Sequences

Arithmetic sequences are characterized by the fact that to get from one term to the next we always add the same amount.
The amount we add is known as the common difference and is usually referred to as \(d\).

For example, the sequence whose first few terms are: \[3,7,11,15,19,23, \dots \] is arithmetic, with common difference \(d = 4\).

Formula for the \(n^{\text{th}}\) term

Given the first few terms of an arithmetic sequence, we can calculate any term of the sequence, using formula: \[u_n = u_1 + \begin{pmatrix} n -1 \end{pmatrix}d \] Where \(u_1\) is the first term of the sequence and \(d\) is its common difference.

This formula can also be rearranged and written: \[u_n = dn+c\] Where \(c\) is a number that we'll quikcly know how to find.

These formula allow us to calculate any term of the sequence directly.

Tutorial

In the following tutorial we review the method for findng the formula for the \(n^{\text{th}}\) term of a linear sequence. Watch it now.

Exercise

  1. Calculate the sum of the first \(20\) terms of the arithmetic sequence whose formula for the \(n^{\text{th}}\) term is: \[u_n = 4n-3\]

  2. Calculate the sum of the first \(30\) terms of the arithmetic sequence defined by the formula: \[u_n = -3n+7 \]

  3. Calculate the sum of the first \(25\) terms of the arithmetic sequence defined by the formula: \[u_n = 7n-2 \]

  4. Given an arithmetic sequence defined by the formula: \[u_n = 6n +5\] Calculate the sum of the \(6^{th}\) to the \(20^{\text{th}}\) terms included.
    Hint: this equals to the difference of two sums.

Answers Without Working

Formula 2

Given an arithmetic sequence, we can calculate the sum of its first \(n\) terms, \(S_n\), using the formula: \[S_n = \frac{n}{2} \begin{pmatrix} 2.u_1 + (n-1)d \end{pmatrix}\] Where \(u_1\) is the first term of the sequence and \(d\) is its common difference.

Note: exam questions frequently involve this second formula.

Exercise 2 - Exam-Type Questions

  1. An arithmetic sequence has twentieth term equal to \(98\) and common difference equal to \(5\).
    Find its first term.

  2. An arithmetic sequence has twelfth term equal to \(-37\) and first term equal to \(7\).
    Find its Difference.

  3. An arithmetic sequence has fifth term equal to \(8\) and twelfth term equal to \(29\).
    Find the values of this sequence's first term, \(u_1\), and its common difference, \(d\).

  4. An arithmetic sequence has tenth term equal to \(43\) and twenty-second term equal to \(91\).
    Find the values of this sequence's first term, \(u_1\), and its common difference, \(d\).

Solutions Without Working

  1. \(u_1 = 3\)

  2. \(d = -4\)

  3. \(u_1 = -4\) and \(d = 3\)

  4. \(u_1 = 7\) and \(d = 4\)