Leaving Cert Mathematics Induction (Proof): (a) Prove by induction that $$

(a) Prove by induction that $$egin{aligned} otag extstyle \ m{ extstyle extstyle extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} m{ extstyle extstyle extstyle} & \ \textstyle extstyle extstyle extstyle\ extstyle extstyle extstyle m{= rac{n(n+1)}{2}} = n(n+1) \ extstyle extstyle extstyle\ onumber \ extstyle n ext{ extbf{and} } n ext{ extbf{and} } n ext{ extbf{ in } } ext{N} - Leaving Cert Mathematics - Question 2 - 2018

Question 2

$(a)-Prove-by-induction-that---$$egin{aligned}-otag-extstyle-\-m{-extstyle--extstyle--extstyle--extstyle--extstyle}-onumberm{-extstyle--extstyle--extstyle}-onumberm{-extstyle--extstyle--extstyle}-m{-extstyle--extstyle--extstyle}-&--\--\textstyle--extstyle-extstyle-extstyle\--extstyle--extstyle-extstyle-m{=--rac{n(n+1)}{2}}-=-n(n+1)-\-extstyle--extstyle-extstyle\-onumber--\--extstyle-n--ext{--extbf{and}-}-n--ext{--extbf{and}-}-n--ext{--extbf{-in-}-}-ext{N}-Leaving Cert Mathematics-Question 2-2018.png$

(a) Prove by induction that $$egin{aligned} otag extstyle \ m{ extstyle extstyle extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumber... show full transcript

Worked Solution & Example Answer:(a) Prove by induction that $$egin{aligned} otag extstyle \ m{ extstyle extstyle extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} m{ extstyle extstyle extstyle} & \ \textstyle extstyle extstyle extstyle\ extstyle extstyle extstyle m{= rac{n(n+1)}{2}} = n(n+1) \ extstyle extstyle extstyle\ onumber \ extstyle n ext{ extbf{and} } n ext{ extbf{and} } n ext{ extbf{ in } } ext{N} - Leaving Cert Mathematics - Question 2 - 2018

Step 1

Prove by induction that $$ extstylem{ extstyle m{ extstyle extstyle extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} = rac{n(n+1)}{2}} ext{ for any } n ext{ in } ext{N}.$$

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Answer

To prove the statement by induction, we first verify the base case. For $n=1$ , the left-hand side is: $extstylem{ extstyle extstyle extstyle rac{1(1+1)}{2} = 1}$ . This matches the right-hand side, proving the base case.

Next, assume the statement is true for some $n=k$ : $extstylem{ extstyle extstyle extstyle rac{k(k+1)}{2}}$ Now we must show it holds for $n=k+1$ :

$extstylem{ extstyle rac{(k+1)(k+2)}{2}} = rac{k(k+1)}{2} + (k+1)$ $= rac{k(k+1) + 2(k+1)}{2} = rac{(k+1)(k+2)}{2}$ This proves the induction step, hence the formula holds for all natural numbers $n$ .

Step 2

State the range of values of x for which the series $$ extstylem{ extstyle extstyle extstyle extstyle extstyle extstyle extstyle(4x-1)^{r}}$$ is convergent.

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Answer

This series is geometric, and its convergence depends on the common ratio: $|4x-1| < 1$ . This leads us to the inequalities: $-1 < 4x - 1 < 1$ , which simplifies to: $0 < x < rac{1}{2}$ .

Now, for the infinite sum, the formula for a converging geometric series is given by: $S_ ext{inf} = rac{a}{1 - r}$ , where $a = 4x-1$ and the common ratio $r = 4x - 1$ must satisfy $|4x-1| < 1$ . Hence, the infinite sum can be expressed as: $S_ ext{inf} = rac{1}{1 - (4x - 1)} = rac{1}{2 - 4x},$ valid for $0 < x < rac{1}{2}$ .

Prove by induction that $$ extstylem{ extstyle m{ extstyle extstyle extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} = rac{n(n+1)}{2}} ext{ for any } n ext{ in } ext{N}.$$

State the range of values of x for which the series $$ extstylem{ extstyle extstyle extstyle extstyle extstyle extstyle extstyle(4x-1)^{r}}$$ is convergent.

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Prove by induction that $$ extstylem{ extstyle m{ extstyle extstyle extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} onumberm{ extstyle extstyle extstyle} = rac{n(n+1)}{2}} ext{ for any } n ext{ in } ext{N}.$$

State the range of values of x for which the series $$ extstylem{ extstyle extstyle extstyle extstyle extstyle extstyle extstyle(4x-1)^{r}}$$ is convergent.