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Chapter 12: Q. 6 (page 830)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n :

1+4+7+...+(3n-2)=12n(3n-1)

Short Answer

Expert verified

The statement is true for alln鈭�N

Step by step solution

01

. Given Information

We are given a statement :

1+4+7+...+(3n-2)=12n(3n-1)

02

. Checking for n=1

P(1)=1(L.H.S)

P(1)=1(3-1)2=1(R.H.S)

Therefore the statement is true for n=1

03

. Checking for n=k

Let P(n) is true for n=k so we need to prove that P(k+1) is true :P(k)=1+4+7+...+(3k-2)=12k(3k-1) -equation(i)

So, the next term will be :

1+4+7+...+(3k-2)+(3k+1)=(k+1)(3k+2)2

Now we need to modify the L.H.S so that it is equal to the R.H.S, we can substitute the value of equation one in this :

k(3k-1)+2(3k+1)2=3k2-k+6k+22

3k2+5k+22=(k+1)(3k+2)2

So, R.H.S = L.H.S which means P(n) is true for n=k+1

Hence , the statement is true for alln鈭�N

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Most popular questions from this chapter

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

a1=3;an=an-1n

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

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

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

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

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