8. Find The First Five Terms Of The Sequence Given By $a_1=2, \, A_n=3a_{n-1}-1$.9. The First Term Of An Arithmetic Sequence Is 2 And The Fourth Term Is 11. Find The Sum Of The First 50 Terms.10. For The Following Questions, Use The System Of

8. Finding the First Five Terms of a Sequence

In this section, we will explore a sequence defined by the recursive formula $a_n=3a_{n-1}-1$, with the initial term $a_1=2$. Our goal is to find the first five terms of this sequence.

Understanding the Recursive Formula

The recursive formula $a_n=3a_{n-1}-1$ indicates that each term in the sequence is obtained by multiplying the previous term by 3 and then subtracting 1. This means that to find the next term in the sequence, we need to apply this formula to the previous term.

Finding the First Five Terms

Let's start by finding the first five terms of the sequence.

• The first term is given as $a_1=2$.
• To find the second term, we apply the recursive formula: $a_2=3a_1-1=3(2)-1=6-1=5$.
• To find the third term, we apply the recursive formula again: $a_3=3a_2-1=3(5)-1=15-1=14$.
• To find the fourth term, we apply the recursive formula once more: $a_4=3a_3-1=3(14)-1=42-1=41$.
• To find the fifth term, we apply the recursive formula one last time: $a_5=3a_4-1=3(41)-1=123-1=122$.

Therefore, the first five terms of the sequence are $2, 5, 14, 41, 122$.

9. Finding the Sum of the First 50 Terms of an Arithmetic Sequence

In this section, we will explore an arithmetic sequence with the first term $a_1=2$ and the fourth term $a_4=11$. Our goal is to find the sum of the first 50 terms of this sequence.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, we are given the first term $a_1=2$ and the fourth term $a_4=11$. We need to find the common difference $d$ between consecutive terms.

Finding the Common Difference

To find the common difference $d$, we can use the formula for the nth term of an arithmetic sequence: $a_n=a_1+(n-1)d$. We know that $a_4=11$, so we can set up the equation:

$$11=2+(4-1)d$$

Simplifying the equation, we get:

$$11=2+3d$$

Subtracting 2 from both sides, we get:

$$9=3d$$

Dividing both sides by 3, we get:

$$d=3$$

Therefore, the common difference between consecutive terms is $d=3$.

Finding the Sum of the First 50 Terms

Now that we have found the common difference $d$, we can use the formula for the sum of the first n terms of an arithmetic sequence: $S_n=\frac{n}{2}(a_1+a_n)$. We know that $n=50$, $a_1=2$, and $a_n=a_1+(n-1)d=a_1+49d=2+49(3)=2+147=149$. Plugging these values into the formula, we get:

$$S_{50}=\frac{50}{2}(2+149)=25(151)=3775$$

Therefore, the sum of the first 50 terms of the arithmetic sequence is $3775$.

10. Solving Systems of Linear Equations

In this section, we will explore a system of linear equations and use it to solve a problem.

Understanding Systems of Linear Equations

A system of linear equations is a set of two or more linear equations that are solved simultaneously. In this case, we will use a system of two linear equations to solve a problem.

Solving the System of Linear Equations

Let's consider the following system of linear equations:

$$x+y=4$$

$$2x-y=1$$

We can solve this system of linear equations using the method of substitution or elimination. Let's use the method of elimination.

Using the Method of Elimination

To use the method of elimination, we need to multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:

$$x+y=4 \quad \text{(Equation 1)}$$

$$4x-y=4 \quad \text{(Equation 2)}$$

Now, we can add both equations to eliminate the y-variable:

$$(x+y)+(4x-y)=4+4$$

Simplifying the equation, we get:

$$5x=8$$

Dividing both sides by 5, we get:

$$x=\frac{8}{5}$$

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's substitute it into Equation 1:

$$\frac{8}{5}+y=4$$

Subtracting $\frac{8}{5}$ from both sides, we get:

$$y=4-\frac{8}{5}$$

Simplifying the expression, we get:

$$y=\frac{20-8}{5}=\frac{12}{5}$$

Therefore, the solution to the system of linear equations is $x=\frac{8}{5}$ and $y=\frac{12}{5}$.

Conclusion

In this article, we have explored three different problems involving sequences and series. We have found the first five terms of a sequence defined by a recursive formula, the sum of the first 50 terms of an arithmetic sequence, and the solution to a system of linear equations. These problems demonstrate the importance of mathematical sequences and series in real-world applications and the need for a deep understanding of these concepts.