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Can somebody prove this mathmatical induction?

Can somebody prove this mathmatical induction?-example-1

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Answer:

See explanation

Explanation:

1 step:

n=1, then


\sum \limits_(j=1)^1 2^j=2^1=2\\ \\2(2^1-1)=2(2-1)=2\cdot 1=2

So, for j=1 this statement is true

2 step:

Assume that for n=k the following statement is true


\sum \limits_(j=1)^k2^j=2(2^k-1)

3 step:

Check for n=k+1 whether the statement


\sum \limits_(j=1)^(k+1)2^j=2(2^(k+1)-1)

is true.

Start with the left side:


\sum \limits _(j=1)^(k+1)2^j=\sum \limits _(j=1)^k2^j+2^(k+1)\ \ (\ast)

According to the 2nd step,


\sum \limits_(j=1)^k2^j=2(2^k-1)

Substitute it into the
\ast


\sum \limits _(j=1)^(k+1)2^j=\sum \limits _(j=1)^k2^j+2^(k+1)=2(2^k-1)+2^(k+1)=2^(k+1)-2+2^(k+1)=2\cdot 2^(k+1)-2=2^(k+2)-2=2(2^(k+1)-1)

So, you have proved the initial statement

User Ivarpoiss
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