1. Solve the equation by using Cramer's rule: x + 2y + 3z = 17; 3x + 2y + z = 11; x - 5y + z = -5. 2. Find the 25th term: 12, 16, 20, 24, ----. 3. Determine the following 3 terms... 1. Solve the equation by using Cramer's rule: x + 2y + 3z = 17; 3x + 2y + z = 11; x - 5y + z = -5. 2. Find the 25th term: 12, 16, 20, 24, ----. 3. Determine the following 3 terms for the Fibonacci Sequence: 0, 2, 2, 4, 6, 10, ----, ----, ----. 4. Determine if the series converges or diverges: Σn=0 to ∞ (2n-1)/(3n). 5. Find the Taylor series for f(x) = ln(1+x), a = 0. Read more

Steps to Solve

1. Solving the System of Equations using Cramer's Rule

To solve the system of equations using Cramer's rule, we first define the equations:

[ \begin{align*}

1. & \quad x + 2y + 3z = 17 \

2. & \quad 3x + 2y + z = 11 \

3. & \quad x - 5y + z = -5 \end{align*} ]

   • The coefficient matrix ( A ) is: $$ A = \begin{pmatrix} 1 & 2 & 3 \ 3 & 2 & 1 \ 1 & -5 & 1 \end{pmatrix} $$

   • The constants matrix ( B ) is: $$ B = \begin{pmatrix} 17 \ 11 \ -5 \end{pmatrix} $$

   • The determinants are calculated as follows:

   • Find the determinant of ( A ) (denoted ( \Delta )): $$ \Delta = \text{det}(A) = 1(21 - 1(-5)) - 2(31 - 1(-5)) + 3(3*(-5) - 2*1) = 1(2+5) - 2(3+5) + 3(-15-2) = 7 - 16 - 51 = -60 $$

   • Find the determinants ( \Delta_x, \Delta_y, \Delta_z ) by replacing columns one by one with ( B ).

2. Finding ( \Delta_x )

Replace the first column of ( A ) with ( B ): $$ \Delta_x = \begin{vmatrix} 17 & 2 & 3 \ 11 & 2 & 1 \ -5 & -5 & 1 \end{vmatrix} $$ Calculate ( \Delta_x ).

3. Finding ( \Delta_y )

Replace the second column of ( A ) with ( B ): $$ \Delta_y = \begin{vmatrix} 1 & 17 & 3 \ 3 & 11 & 1 \ 1 & -5 & 1 \end{vmatrix} $$ Calculate ( \Delta_y ).

4. Finding ( \Delta_z )

Replace the third column of ( A ) with ( B ): $$ \Delta_z = \begin{vmatrix} 1 & 2 & 17 \ 3 & 2 & 11 \ 1 & -5 & -5 \end{vmatrix} $$ Calculate ( \Delta_z ).

5. Calculate values of ( x, y, z )

Using Cramer’s Rule: $$ x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta} $$

6. Finding the 25th term in the sequence

The given sequence is an arithmetic sequence: ( 12, 16, 20, 24, \ldots )

• The common difference ( d = 16 - 12 = 4 ).

• The formula for the nth term of an arithmetic sequence is: $$ a_n = a_1 + (n-1)d $$ where ( a_1 = 12 ) and ( d = 4 ).

• Thus, find ( a_{25} ): $$ a_{25} = 12 + (25 - 1) \cdot 4 $$

7. Finding the next 3 terms in the Fibonacci Sequence

The Fibonacci sequence begins: ( 0, 1, 1, 2, 3, 5, \ldots )

• The next terms are calculated as:
• ( F_3 = F_1 + F_2 )
• Continue till three terms are found: ( \ldots, 13, 21, 34 ).

8. Determining convergence or divergence of the series

The series to evaluate is: $$ \sum_{n=0}^{\infty} \frac{2^{n-1}}{3^n} $$

• Use the ratio test, where if the limit of the ratio of successive terms is less than 1, the series converges.

9. Finding the Taylor series expansion for ( f(x) = \ln(1 + x) )

The Taylor series about ( a = 0 ): $$ f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} $$