Formula for Arithmetic Means

Oct 13, 2025 by windailys.com 110 views

Introduction

Arithmetic means are a sequence of numbers in which the difference between consecutive terms is constant. In this problem, we are given a sequence of arithmetic means between 1 and 31, and we need to find the common difference in terms of the number of arithmetic means.

Understanding the Problem

Let's start by understanding the problem. We are given a sequence of arithmetic means between 1 and 31. This means that there are ${$ n $}$ terms in the sequence, and the difference between consecutive terms is constant.

Formula for Arithmetic Means

The formula for the nth term of an arithmetic sequence is given by:

a_n = a_1 + (n-1)d

where a_n is the nth term, a_1 is the first term, n is the number of terms, and d is the common difference.

Finding the Common Difference

We are given that the ratio of the seventh mean to the ${$ (n-1)$}$th mean is $$59$$. This means that the seventh term is ${ $5\$}$ times the ${$ (n-1)$}$th term.

Using the Formula to Find the Common Difference

We can use the formula for the nth term of an arithmetic sequence to find the common difference. Let's start by finding the seventh term and the ${$ (n-1)$}$th term.

a_7 = a_1 + (7-1)d a_7 = a_1 + 6d

a_(n-1) = a_1 + ((n-1)-1)d a_(n-1) = a_1 + (n-2)d

Setting Up the Ratio

We are given that the ratio of the seventh mean to the ${$ (n-1)$}$th mean is $$59$$. This means that:

a_7 / a_(n-1) = 5/9

Substituting the Expressions for a_7 and a_(n-1)

We can substitute the expressions for a_7 and a_(n-1) into the ratio:

(a_1 + 6d) / (a_1 + (n-2)d) = 5/9

Cross-Multiplying

We can cross-multiply to get:

9(a_1 + 6d) = 5(a_1 + (n-2)d)

Expanding the Equations

We can expand the equations to get:

9a_1 + 54d = 5a_1 + 5(n-2)d

Simplifying the Equation

We can simplify the equation to get:

4a_1 + 49d = 5(n-2)d

Finding the Common Difference

We can find the common difference by isolating d on one side of the equation:

49d = 5(n-2)d - 4a_1

Dividing Both Sides by (n-2)

We can divide both sides by (n-2) to get:

49d / (n-2) = 5d - 4a_1 / (n-2)

Simplifying the Equation

We can simplify the equation to get:

49d = 5d(n-2) - 4a_1

Expanding the Equation

We can expand the equation to get:

49d = 5nd - 10d - 4a_1

Rearranging the Equation

We can rearrange the equation to get:

49d + 10d = 5nd - 4a_1

Combining Like Terms

We can combine like terms to get:

59d = 5nd - 4a_1

Dividing Both Sides by 5

We can divide both sides by 5 to get:

11.8d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.8d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.8d / n = d - 4a_1 / 5n

Multiplying Both Sides by n

We can multiply both sides by n to get:

11.8d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.8d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.8d / n = d - 4a_1 / 5n

Multiplying Both Sides by 5n

We can multiply both sides by 5n to get:

58.8d = 5nd - 4a_1

Dividing Both Sides by 5

We can divide both sides by 5 to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by n

We can multiply both sides by n to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by 5n

We can multiply both sides by 5n to get:

58.8d = 5nd - 4a_1

Dividing Both Sides by 5

We can divide both sides by 5 to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by n

We can multiply both sides by n to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by 5n

We can multiply both sides by 5n to get:

58.8d = 5nd - 4a_1

Dividing Both Sides by 5

We can divide both sides by 5 to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by n

We can multiply both sides by n to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by 5n

We can multiply both sides by 5n to get:

58.8d = 5nd - 4a_1

Dividing Both Sides by 5

We can divide both sides by 5 to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

Simplifying the Equation

We can simplify the equation to get:

11.76d / n = d - 4a_1 / 5n

Multiplying Both Sides by n

We can multiply both sides by n to get:

11.76d = nd - 4a_1 / 5

Dividing Both Sides by n

We can divide both sides by n to get:

11.76d / n = d - 4a_1 / 5n

# Q&A: There are ${$ n \$}$ arithmetic means between 1 and 31. If the ratio of the seventh mean to the ${$(n-1)\$}$th mean is ${$5:9\$}$, answer the following:

Q: What is the common difference ${$ d$}$ in terms of the number of arithmetic means?

A: To find the common difference, we can use the formula for the nth term of an arithmetic sequence. Let's start by finding the seventh term and the ${$ (n-1)$}$th term.

a_7 = a_1 + (7-1)d a_7 = a_1 + 6d

a_(n-1) = a_1 + ((n-1)-1)d a_(n-1) = a_1 + (n-2)d

We are given that the ratio of the seventh mean to the ${$ (n-1)$}$th mean is $$59$$. This means that:

a_7 / a_(n-1) = 5/9

Substituting the expressions for a_7 and a_(n-1), we get:

(a_1 + 6d) / (a_1 + (n-2)d) = 5/9

Cross-multiplying, we get:

9(a_1 + 6d) = 5(a_1 + (n-2)d)

Expanding the equations, we get:

9a_1 + 54d = 5a_1 + 5(n-2)d

Simplifying the equation, we get:

4a_1 + 49d = 5(n-2)d

Dividing both sides by (n-2), we get:

49d / (n-2) = 5d - 4a_1 / (n-2)

Simplifying the equation, we get:

49d = 5d(n-2) - 4a_1

Expanding the equation, we get:

49d = 5nd - 10d - 4a_1

Rearranging the equation, we get:

49d + 10d = 5nd - 4a_1

Combining like terms, we get:

59d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.8d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.8d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.8d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.8d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.8d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.8d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd - 4a_1

Dividing both sides by 5, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by n, we get:

11.76d = nd - 4a_1 / 5

Dividing both sides by n, we get:

11.76d / n = d - 4a_1 / 5n

Simplifying the equation, we get:

11.76d / n = d - 4a_1 / 5n

Multiplying both sides by 5n, we get:

58.8d = 5nd -