Chapter 12: Q 105. (page 812)

For the sequence given in Problem 103, show that
$u_{n + 1} + u_{n} = {(n + 1)}^{2}$

Short Answer

Expert verified

It is shown that $u_{n + 1} + u_{n} = {(n + 1)}^{2}$ .

Step by step solution

Step 1. Write the given information.

The given information is:

$u_{1} = 1; u_{n + 1} = u_{n} + (n + 1)$

We have to show that $u_{n + 1} + u_{n} = {(n + 1)}^{2}$

Step 2. Find a few terms with the help of the given formula.

$u_{1} = 1 u_{2} = u_{1} + (1 + 1) = 1 + 2 = 3 u_{3} = u_{2} + (2 + 1) = 3 + 3 = 6 u_{n} = u_{n - 1} + (n - 1 + 1) u_{n - 1} = u_{n - 2} + (n - 2 + 1)$

Step 3. Substitute the above-calculated values in the formula un+1=un+n+1.

^{$u_{n + 1} = u_{n} + (n + 1) = (u_{n - 1} + (n - 1 + 1)) + (n + 1) = (u_{n - 2} + (n - 2 + 1)) + n + (n + 1) = u_{n - 3} + (n - 3 + 1) + (n - 1) + n + (n + 1)$}

On continuing the series in the same way we will get the following series,

$u_{n + 1} = u_{1} + (2 + 3 + . . . . + (n + 1)) = 1 + 2 + 3 + . . . . (n + 1)$

The formula of first n natural numbers is: $1 + 2 + 3 + . . . . (n + 1) = \frac{n (n + 1)}{2}$

Therefore, $u_{n + 1} = \frac{(n + 1) (n + 2)}{2}$

Similarly $u_{n} = \frac{n (n + 1)}{2}$

Step 4. Find un+1+un.

$u_{n + 1} + u_{n} = \frac{(n + 1) (n + 2)}{2} + \frac{n (n + 1)}{2} = \frac{(n + 1) (n + 2) + n (n + 1)}{2} = \frac{n^{2} + n + n^{2} + 2 n + n + 2}{2} = \frac{2 n^{2} + 4 n + 2}{2} = n^{2} + 2 n + 1 = {(n + 1)}^{2}$

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For the sequence given in Problem 103, show that
$u_{n + 1} + u_{n} = {(n + 1)}^{2}$

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Find a few terms with the help of the given formula.

Step 3. Substitute the above-calculated values in the formula un+1=un+n+1.

Step 4. Find un+1+un.

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For the sequence given in Problem 103, show thatun+1+un=n+12

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Find a few terms with the help of the given formula.

Step 3. Substitute the above-calculated values in the formula un+1=un+n+1.

Step 4. Find un+1+un.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

For the sequence given in Problem 103, show that
$u_{n + 1} + u_{n} = {(n + 1)}^{2}$