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Chapter 12: Q. 27 (page 839)

In Problems 26鈥�28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.

2+6+18+................2.3n-1=3n-1

Short Answer

Expert verified

Given statement is true for all natural numbers.

Step by step solution

01

Step 1. Given information. 

Statement:

2+6+18+...........2.3n-1=3n-1

02

Step 2. Show that statement is true for n=1.

L.H.S.=2

R.H.S.=31-1=3-1=2

L.H.S=R.H.S.

Statement is true.

03

Step 3. Assume that statement is true for n=k

2+6+18+.................2.3k-1=3k-1

04

Step 4. Show that the statement is true for n=k+1.

L.H.S.=2+6+18+.......+2.3k-1+2.3(K+1)-1=3k-1+2.3(K+1)-1=3k-1+2.3k=3k(1+2)-1=3k.31-1=3k+1-1, use result of step 3.

R.H.S.=3K+1-1

Since L.H.S.=R.H.S.for n=k+1

So statement is true for all natural number.

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