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Chapter 12: Q. 26 (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.

3+6+9+12+........3n=3n2n+1

Short Answer

Expert verified

Given statement is true for all natural numbers.

Step by step solution

01

Step 1. Given information.

Statement:

3+6+9+12+.......+3n=3n2(n+1)

02

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

L.H.S=3

R.H.S.=3脳12(1+1)=32脳2=3

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

Statement is true.

03

Step 3. Assume that statement is true for k.

3+6+9+..........+3k=3k2(k+1)

04

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

L.H.S.=3+6+9+....+3k+3(k+1)=3k2(k+1)+3(k+1)=3(k+1)k2+1=3(k+1)k+22=32(k+1)(k+2), use result of step 3.

R.H.S.=3(k+1)2(k+1+1)=3(k+1)2(k+2)

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

Hence the statement is true for all natural number

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