If A 1 , A 2 , A 3 A_1,a_2,a_3 A 1 ​ , A 2 ​ , A 3 ​ Is Geometric Sequence Such That A 1 + A 2 + A 3 = 91 A_1+a_2+a_3=91 A 1 ​ + A 2 ​ + A 3 ​ = 91 And A 1 , A 2 , ( A 3 − 13 ) A_1, A_2, (a_3-13) A 1 ​ , A 2 ​ , ( A 3 ​ − 13 ) Is Arithmetic Sequence, What The Value Of A 1 A_1 A 1 ​ ?

Introduction

Sequences and series are fundamental concepts in mathematics, and understanding their properties is crucial for solving various problems in algebra, geometry, and calculus. In this article, we will delve into the world of geometric and arithmetic sequences, exploring their definitions, properties, and applications. Specifically, we will focus on a problem that involves both geometric and arithmetic sequences, and we will use mathematical reasoning and problem-solving techniques to find the value of the first term of the geometric sequence.

What are Geometric and Arithmetic Sequences?

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In other words, if $a_1, a_2, a_3, \ldots$ is a geometric sequence, then $a_2 = a_1 \cdot r$, $a_3 = a_2 \cdot r = a_1 \cdot r^2$, and so on, where $r$ is the common ratio.

On the other hand, an arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed number called the common difference to the previous term. In other words, if $a_1, a_2, a_3, \ldots$ is an arithmetic sequence, then $a_2 = a_1 + d$, $a_3 = a_2 + d = a_1 + 2d$, and so on, where $d$ is the common difference.

The Problem

We are given that $a_1, a_2, a_3$ is a geometric sequence such that $a_1 + a_2 + a_3 = 91$, and $a_1, a_2, (a_3-13)$ is an arithmetic sequence. Our goal is to find the value of $a_1$.

Step 1: Write Down the Equations

Since $a_1, a_2, a_3$ is a geometric sequence, we can write:

$$a_2 = a_1 \cdot r$$

$$a_3 = a_2 \cdot r = a_1 \cdot r^2$$

where $r$ is the common ratio.

Since $a_1, a_2, (a_3-13)$ is an arithmetic sequence, we can write:

$$a_2 = a_1 + d$$

$$(a_3-13) = a_2 + d = a_1 + 2d$$

where $d$ is the common difference.

Step 2: Substitute the Expressions for $a_2$ and $a_3$

We can substitute the expressions for $a_2$ and $a_3$ into the equation $a_1 + a_2 + a_3 = 91$:

$$a_1 + (a_1 \cdot r) + (a_1 \cdot r^2) = 91$$

We can also substitute the expression for $a_2$ into the equation $(a_3-13) = a_1 + 2d$:

$$(a_1 \cdot r^2 - 13) = a_1 + 2d$$

Step 3: Simplify the Equations**

We can simplify the equation $a_1 + (a_1 \cdot r) + (a_1 \cdot r^2) = 91$ by factoring out $a_1$:

$$a_1(1 + r + r^2) = 91$$

We can also simplify the equation $(a_1 \cdot r^2 - 13) = a_1 + 2d$ by rearranging the terms:

$$a_1 \cdot r^2 - a_1 - 2d = 13$$

Step 4: Solve for $a_1$

We can solve for $a_1$ by using the equation $a_1(1 + r + r^2) = 91$. We can divide both sides of the equation by $(1 + r + r^2)$:

$$a_1 = \frac{91}{1 + r + r^2}$$

We can also solve for $a_1$ by using the equation $a_1 \cdot r^2 - a_1 - 2d = 13$. We can rearrange the terms to get:

$$a_1 \cdot r^2 - a_1 = 2d + 13$$

We can factor out $a_1$ from the left-hand side of the equation:

$$a_1(r^2 - 1) = 2d + 13$$

We can divide both sides of the equation by $(r^2 - 1)$:

$$a_1 = \frac{2d + 13}{r^2 - 1}$$

Step 5: Equate the Two Expressions for $a_1$

We can equate the two expressions for $a_1$:

$$\frac{91}{1 + r + r^2} = \frac{2d + 13}{r^2 - 1}$$

We can cross-multiply to get:

$$(91)(r^2 - 1) = (2d + 13)(1 + r + r^2)$$

We can expand the right-hand side of the equation:

$$(91)(r^2 - 1) = 2d + 13 + 2dr + 13r + 13r^2$$

We can simplify the equation by combining like terms:

$$(91)(r^2 - 1) = 2d + 13 + 2dr + 13r + 13r^2$$

Step 6: Solve for $r$ and $d$

We can solve for $r$ and $d$ by using the equation $(91)(r^2 - 1) = 2d + 13 + 2dr + 13r + 13r^2$. We can rearrange the terms to get:

$$(91r^2 - 91) = 2d + 13 + 2dr + 13r + 13r^2$$

We can simplify the equation by combining like terms:

$$(91r^2 - 13r^2) - 91 = 2d + 13 + 2dr + 13r$$

We can factor out $r^2$ from the left-hand side of the equation:

$$(78r^2 - 13r) - 91 = 2d + 13 + 2dr + 13r$$

We can simplify the equation by combining like terms$(78r^2 - 13r - 91) = 2d + 13 + 2dr + 13r$

We can rearrange the terms to get:

$$(78r^2 - 13r - 91) - (2d + 13 + 2dr + 13r) = 0$$

We can simplify the equation by combining like terms:

$$(78r^2 - 13r - 91) - (2d + 13 + 2dr + 13r) = 0$$

We can factor out $r$ from the left-hand side of the equation:

$$(78r^2 - 26r - 91) - (2d + 13) = 0$$

We can simplify the equation by combining like terms:

$$(78r^2 - 26r - 91) - (2d + 13) = 0$$

We can factor out $r$ from the left-hand side of the equation:

$$(78r^2 - 26r - 104) = 2d$$

We can simplify the equation by combining like terms:

$$(78r^2 - 26r - 104) = 2d$$

We can divide both sides of the equation by $2$:

$$(39r^2 - 13r - 52) = d$$

Step 7: Find the Value of $a_1$

We can find the value of $a_1$ by substituting the expression for $d$ into the equation $a_1 = \frac{91}{1 + r + r^2}$. We can get:

$$a_1 = \frac{91}{1 + r + r^2}$$

We can substitute the expression for $d$ into the equation $a_1 = \frac{2d + 13}{r^2 - 1}$. We can get:

$$a_1 = \frac{2(39r^2 - 13r - 52) + 13}{r^2 - 1}$$

We can simplify the equation by combining like terms:

$$a_1 = \frac{78r^2 - 26r - 104 + 13}{r^2 - 1}$$

We can simplify the equation by combining like terms:

$$a_1 = \frac{78r^2 - 26r - 91}{r^2 - 1}$$

We can factor out $r$ from the numerator of the equation:

a_1 = \frac{r(78r -<br/> **Q&A: Unraveling the Mystery of Geometric and Arithmetic Sequences** ====================================================================

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed number called the common difference to the previous term.

Q: How do we find the value of $a_1$ in a geometric sequence?

A: To find the value of $a_1$ in a geometric sequence, we can use the formula $a_1 = \frac{91}{1 + r + r^2}$, where $r$ is the common ratio.

Q: How do we find the value of $a_1$ in an arithmetic sequence?

A: To find the value of $a_1$ in an arithmetic sequence, we can use the formula $a_1 = \frac{2d + 13}{r^2 - 1}$, where $d$ is the common difference and $r$ is the common ratio.

Q: What is the relationship between geometric and arithmetic sequences?

A: Geometric and arithmetic sequences are related in that they can be used to model real-world phenomena. For example, a geometric sequence can be used to model population growth, while an arithmetic sequence can be used to model the cost of goods over time.

Q: How do we solve for $r$ and $d$ in a geometric and arithmetic sequence?

A: To solve for $r$ and $d$ in a geometric and arithmetic sequence, we can use the equations $(78r^2 - 13r - 91) = 2d + 13 + 2dr + 13r$ and $(39r^2 - 13r - 52) = d$. We can then substitute the expression for $d$ into the equation $a_1 = \frac{91}{1 + r + r^2}$ to find the value of $a_1$.

Q: What are some real-world applications of geometric and arithmetic sequences?

A: Geometric and arithmetic sequences have many real-world applications, including:

• Modeling population growth
• Modeling the cost of goods over time
• Modeling the spread of diseases
• Modeling the growth of investments
• Modeling the decay of radioactive materials

Q: How do we use geometric and arithmetic sequences in finance?

A: Geometric and arithmetic sequences are used in finance to model the growth of investments, the decay of bonds, and the cost of goods over time. For example, a geometric sequence can be used to model the growth of a stock portfolio, while an arithmetic sequence can be used to model the cost of goods over time.

Q: How do we use geometric and arithmetic sequences in science?

A: Geometric and arithmetic sequences are used in science to model the spread of diseases, the growth of populations, and the decay of radioactive materials. For example, a geometric sequence can be used to model the spread of a disease, while an arithmetic sequence can be used to model the growth of a population.

Q: What are some common mistakes to avoid when working with geometric and arithmetic sequences?

A: Some common mistakes to avoid when working with geometric and arithmetic sequences include:

• Not checking for extraneous solutions
• Not using the correct formula for the sequence
• Not checking for errors in the calculations
• Not using the correct units for the variables

Q: How do we check for extraneous solutions in geometric and arithmetic sequences?

A: To check for extraneous solutions in geometric and arithmetic sequences, we can use the following steps:

• Check if the solution is a real number
• Check if the solution is a positive number
• Check if the solution is a rational number
• Check if the solution satisfies the original equation

Q: How do we use technology to solve geometric and arithmetic sequences?

A: Technology can be used to solve geometric and arithmetic sequences by using software or calculators to perform the calculations. For example, a graphing calculator can be used to graph the sequence and find the value of $a_1$.