Problem 55 - Precalculus: Mathematical for Calculus

Question

An arithmetic sequence has first term \(a=5\) and common difference \(d=2 .\) How many terms of this sequence must be added to get \(2700 ?\)

Step-by-Step Solution

Answer

50 terms.

Step 1: Understand the problem

We need to find how many terms, denoted as \(n\), of an arithmetic sequence are needed to obtain a sum of 2700. The first term \(a\) is 5, and the common difference \(d\) is 2.

Step 2: Recall the formula for the sum of an arithmetic sequence

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by:\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \] where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

Step 3: Substitute known values into the sum formula

Substitute \(a = 5\), \(d = 2\), and \(S_n = 2700\) into the formula:\[ 2700 = \frac{n}{2} \times (2 \times 5 + (n-1) \times 2) \] Simplifying inside the parentheses gives:\[ 2700 = \frac{n}{2} \times (10 + 2n - 2) \]\[ 2700 = \frac{n}{2} \times (8 + 2n) \]

Step 4: Simplify and solve the equation for n

Simplify the equation:\[ 2700 = \frac{n}{2} \times 2(n + 4) \]\[ 2700 = n(n + 4) \]Now, rearrange the equation:\[ n^2 + 4n - 2700 = 0 \]

Step 5: Solve the quadratic equation

Use the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 4\), and \(c = -2700\):\[ n = \frac{-4 \pm \sqrt{4^2 - 4 \times 1 \times (-2700)}}{2 \times 1} \]\[ n = \frac{-4 \pm \sqrt{16 + 10800}}{2} \]\[ n = \frac{-4 \pm \sqrt{10816}}{2} \]\[ n = \frac{-4 \pm 104}{2} \]

Step 6: Find feasible n

Calculate both possible values for \(n\):\[ n = \frac{-4 + 104}{2} = 50 \]\[ n = \frac{-4 - 104}{2} = -54 \]Since \(n\) must be a positive integer, the number of terms is \(n = 50\).

Key Concepts

Arithmetic Sequence Sum

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. The sum of the first \( n \) terms in an arithmetic sequence can be calculated using the formula:

• \( S_n = \frac{n}{2} \times (2a + (n-1)d) \)

Here, \( S_n \) is the sum of the sequence, \( n \) is the number of terms, \( a \) is the first term, and \( d \) is the common difference.This formula can also be rewritten as:

• \( S_n = \frac{n}{2} \times (a + l) \)

where \( l \) is the last term of the sequence. Both forms are interchangeable depending on what information you have.
Using this formula is essential when you want to find the total of elements in a sequence like determining how many terms of a sequence need adding to result in a certain sum. Substitute the known values, solve the equation, and you'll determine the required number of terms.

Quadratic Formula

The quadratic formula is a tool used to solve quadratic equations, which take the form \( ax^2 + bx + c = 0 \). The solution for \( x \) is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In an arithmetic sequence sum scenario where solving for \( n \), applying the quadratic formula becomes necessary when deriving \( n \) from an equation like \( n^2 + 4n - 2700 = 0 \).It's essential to understand that the quadratic formula provides two potential solutions, as indicated by the "\( \pm \)" symbol. Evaluate both results, and choose the solution that logically fits the context, usually the positive value, as seen in arithmetic sequence problems.

Common Difference

The common difference \( d \) is a fundamental part of an arithmetic sequence. It is the difference between any two successive terms.For instance, if the sequence is \( 5, 7, 9, ... \), the common difference \( d \) is 2. You find it by subtracting any term from the one that follows it.

• example: \( d = 7 - 5 = 2 \)

Knowing the common difference isn't just about creating the sequence; it’s also crucial for using the sum formula correctly. It helps in simplifying the formula and resolving equations when calculating the sum or locating the number of terms required to reach a specific total.

Sequence Terms

Understanding the elements or terms in a sequence is vital. In arithmetic sequences, we define terms in relation to the first term \( a \) and the common difference \( d \). The general term \( T_n \) is given by:

• \( T_n = a + (n-1) \times d \)

This formula helps in identifying the value of any term position \( n \) in the sequence. If you know the position, you can calculate the term by substituting the sequence's first term and common difference into the formula.Terms are the building blocks of a sequence, and their understanding is foundational when dealing with sequential arithmetic operations such as summing a finite number of sequence elements or finding the position of a specific term.