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Chapter 12: Q. 10 (page 830)

In Problems 1鈥�22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

1+5+52+...+5n-1=145n-1

Short Answer

Expert verified

The statement1+5+52+...+5n-1=145n-1is true for all natural numbers.

Step by step solution

01

Step 1. Given information

The given statement is1+5+52+...+5n-1=145n-1.We need to prove that the given statement is true for all natural numbers by using mathematical induction.

02

Step 2. Proof

Let's prove that the statement is true for n=1.

51-1=1451-1.

50=144.

1=1.

So, it is true for n=1.

Let's prove that the statement is true for n=k.

1+5+52+...+5k-1=145k-1.

Let's prove that the statement is true for n=k+1.

1+5+52+...+5k-1+5k+1-1.

=145k-1+5k.

=545k-14.

=145k+1-1.

The statement is true for n=k+1.

Since the given statement is true for n=1and if the statement is true for some n=kand it is also true for the next natural number n=k+1.

Then by the principle of natural number, the statement is true for all natural numbers.

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Most popular questions from this chapter

In Problems 61鈥�70, express each sum using summation notation.

1e+2e2+3e3+.........+nen

True or False. The notation a5represents the fifth term of a sequence.

In Problems 17鈥�28, write down the first five terms of each sequence.

dn=(-1)n-1n2n-1

Environmental Control: The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutants as a result of industrial waste and that 10% of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake at the end of each year is given by the recursively defined sequence

p0=250;pn=0.9pn-1+15

(a) Determine the amount of pollutant in the lake at the

end of the second year. That is, determine p2 .

(b) Using a graphing utility, provide pollutant amounts for

the next 20 years.

(c) What is the equilibrium level of pollution in Maple

Lake? That is, what is localid="1646823556776" limn鈫�鈭�pn?

Find x so that x, x + 2, and x + 3 are consecutive terms of a geometric sequence.
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