Problem 35 - Algebra and Trigonometry

Question

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 15th term is 0 ; 40th term is -50

Step-by-Step Solution

Answer

First term: 28, common difference: -2, recursive formula: \( a_1 = 28 \), \( a_n = a_{n-1} - 2 \), nth term formula: \( a_n = 30 - 2n \)

Step 1: Identify the terms and positions

Given: 15th term is 0 and 40th term is -50. Let these terms be represented as follows:\[ a_{15} = 0 \]\[ a_{40} = -50 \]

Step 2: Formulate equations for the given terms

Using the general form of an arithmetic sequence, \(a_n = a + (n-1)d\), create equations for the given terms:1. \[ a + 14d = 0 \] (for the 15th term)2. \[ a + 39d = -50 \] (for the 40th term)

Step 3: Solve the system of equations

Subtract the first equation from the second to eliminate \( a \):\[ (a + 39d) - (a + 14d) = -50 - 0 \]\[ 25d = -50 \]\[ d = -2 \]

Step 4: Find the first term

Substitute the common difference \( d \) back into one of the original equations to find \( a \):\[ a + 14(-2) = 0 \]\[ a - 28 = 0 \]\[ a = 28 \]

Step 5: Write the recursive formula

The recursive formula for the sequence is given by:\[ a_1 = 28 \]\[ a_n = a_{n-1} - 2 \] for \( n > 1 \)

Step 6: Derive the formula for the nth term

The formula for the nth term is:\[ a_n = 28 + (n-1)(-2) \]\[ a_n = 28 - 2(n-1) \]\[ a_n = 30 - 2n \]

Key Concepts

Common Difference

The common difference is a fundamental concept in arithmetic sequences. It's the constant amount that each term in the sequence changes from the previous term. This difference is key to forming and understanding the sequence.
In our given problem, we started with:

• 15th term ( a_15 ) is 0
• 40th term ( a_40 ) is -50

Using the common formula for arithmetic sequences, a_n = a + (n-1)d we formed two equations:

• a + 14d = 0 (for 15th term)
• a + 39d = -50 (for 40th term)

By subtracting these equations, we simplified to find the common difference:
a + 39d - (a + 14d) = -50 25d = -50 d = -2. This 'd' value tells us that each term in the sequence decreases by 2 from the previous term.

Recursive Formula

The recursive formula allows you to find any term in an arithmetic sequence if you know the previous term. It expresses each term as a function of the term before it.
For our sequence, the first term (a) is 28. We've already found the common difference (d) to be -2. Plugging these into the recursive formula, we get:
a_1 = 28a_n = a_{n-1} - 2 for n > 1

• a_2 = a_1 - 2 = 28 - 2 = 26
• a_3 = a_2 - 2 = 26 - 2 = 24

And so on... This formula lets you generate each subsequent term by subtracting 2 from the previous term. This is helpful if you're building the sequence step-by-step.

Nth Term Formula

The nth term formula gives you a direct way to find any term in the sequence without computing all the previous ones. Using the initial term (a = 28) and the common difference ( d = -2), we derived the nth term formula.
The general form is:
a_n = a + (n-1)dPlugging in our values:
a_n = 28 + (n-1)(-2).Simplify the equation:
28 + -2n + 2So,
a_n = 30 - 2n.This formula tells you exactly what the value of any term in the sequence is, based on its position (n). For example, for the 5th term (a_5):
a_5 = 30 - 2*5 = 20 Overall, knowing this formula is very powerful. It allows you to quickly find any term in your arithmetic sequence.