Find two numbers such that their sum is $-1$ and the sum of their squares is $61$ .

Find examples to show that if the series $\sum u_k$ and $\sum v_k$ both diverge, then the series $\sum\left(u_k+v_k\right)$ and $\sum\left(u_k-v_k\right)$ may either converge or diverge.

Find the sum. Find the sum of the numbers from 51 to 500.

Write the sum of 63 and 15 as the product of their GCF and another sum.

Find the sum.

$$\sum _ { i = 1 } ^ { 5 } 7 i$$

Find the sum.

$$\sum _ { i = 1 } ^ { 6 } 2 i$$

For each initial-value problem, use the Runge-Kutta method and a calculator to approximate the values of the exact solution at each given $x .$ Obtain the exact solution $\phi$ and evaluate it at each $x .$ Compare the approximations to the exact values by calculating the errors and percentage relative errors.

$$y^{\prime}=\sin 2 x+y, \quad y(0)=1$$

Approximate $\phi$ at $x=0.25,0.5, \ldots, 2.0 .$ $(h=0.25)$

Write the sum of 120 and 36 as the product of their GCF and another sum.

Write the sum of 24 and 40 as the product of their GCF and another sum.

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

$$\sum _ { i = 1 } ^ { 30 } ( - 5 i )$$

Write the sum of 18 and 27 as the product of their GCF and another sum.

Find the sum. Find the sum of the numbers from 1949 to 2006.

Write the sum of 32 and 20 as the product of their GCF and another sum.

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.

$$\sum _ { i = 1 } ^ { 7 } 3 ( - 2 ) ^ { i }$$