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Chapter 12: Q 104. (page 812)

For the sequence given in Problem 103, show thatun+1=n+1n+22.

Short Answer

Expert verified

It is shown that un+1=n+1n+22.

Step by step solution

01

Step 1. Write the given information.

The given information is:

u1=1;un+1=un+n+1

We have to show thatun+1=n+1n+22.

02

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

u1=1u2=u1+1+1=1+2=3u3=u2+2+1=3+3=6un=un-1+n-1+1un-1=un-2+n-2+1

03

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

un+1=un+n+1=un-1+n-1+1+n+1=un-2+n-2+1+n+n+1=un-3+n-3+1+n-1+n+n+1

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

un+1=u1+2+3+....+n+1=1+2+3+....n+1

The formula of first n natural numbers is: role="math" localid="1646929459241" 1+2+3+....n+1=nn+12

Therefore,un+1=n+1n+22

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Most popular questions from this chapter

In Problems 71-82, find the sum of each sequence.

鈭�k3k=520

In Problems 11鈥�16, evaluate each factorial expression.

10!

Fibonacci Sequence: Let

un=(1+5)n-(1-5)n2n5

define the nth term of a sequence.

(a) Show that u1= 1 and u2 = 1.

(b) Show that un+2 = un+1 + un.

(c) Draw the conclusion that {un} is the Fibonacci sequence.

In Problems 51鈥�60, write out each sum.

鈭�k=0n32k

In Problems 29鈥�36, the given pattern continues. Write down the nth term of a sequence ansuggested by the pattern.

role="math" localid="1646738671855" 11路2,12路3,13路4,14路5,...
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