Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 109729 by lazygorilla last updated on 25/Aug/20

Answered by prakash jain last updated on 25/Aug/20

1+x+x^2 +..+x^(n−1) =(((x^n −1))/(x−1)) (x≠1) (Geometric Progression) 2^0 +2^1 +...+2^(m−1) =2^m −1 substitute this and you will get the result

$$\mathrm{1}+{x}+{x}^{\mathrm{2}} +..+{x}^{{n}−\mathrm{1}} =\frac{\left({x}^{{n}} −\mathrm{1}\right)}{{x}−\mathrm{1}}\:\:\:\left({x}\neq\mathrm{1}\right) \\ $$$$\:\:\:\:\left(\mathrm{Geometric}\:\mathrm{Progression}\right) \\ $$$$\mathrm{2}^{\mathrm{0}} +\mathrm{2}^{\mathrm{1}} +...+\mathrm{2}^{{m}−\mathrm{1}} =\mathrm{2}^{{m}} −\mathrm{1} \\ $$$$\mathrm{substitute}\:\mathrm{this}\:\mathrm{and}\:\mathrm{you}\:\mathrm{will}\:\mathrm{get}\:\mathrm{the} \\ $$$$\mathrm{result} \\ $$

Answered by bobhans last updated on 25/Aug/20

consider GP : 1+2+4+8+...+2^(m−1) = ((1(2^m −1))/(2−1)) = 2^m −1

$${consider}\:{GP}\::\:\mathrm{1}+\mathrm{2}+\mathrm{4}+\mathrm{8}+...+\mathrm{2}^{{m}−\mathrm{1}} \:=\:\frac{\mathrm{1}\left(\mathrm{2}^{{m}} −\mathrm{1}\right)}{\mathrm{2}−\mathrm{1}} \\ $$$$=\:\mathrm{2}^{{m}} −\mathrm{1}\: \\ $$

Answered by 1549442205PVT last updated on 25/Aug/20

2^0 +2^1 +2^2 +....+2^(m−1) is the sum of m terms of geometric progress with first term be u_1 =2^0 =1,q=2,so S=u_1 ((q^m −1)/(q−1))=1.((2^m −1)/(2−1))=2^m −1

$$\mathrm{2}^{\mathrm{0}} +\mathrm{2}^{\mathrm{1}} +\mathrm{2}^{\mathrm{2}} +....+\mathrm{2}^{\mathrm{m}−\mathrm{1}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{m}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{geometric}\:\mathrm{progress}\:\mathrm{with} \\ $$$$\mathrm{first}\:\mathrm{term}\:\mathrm{be}\:\mathrm{u}_{\mathrm{1}} =\mathrm{2}^{\mathrm{0}} =\mathrm{1},\mathrm{q}=\mathrm{2},\mathrm{so} \\ $$$$\mathrm{S}=\mathrm{u}_{\mathrm{1}} \frac{\mathrm{q}^{\mathrm{m}} −\mathrm{1}}{\mathrm{q}−\mathrm{1}}=\mathrm{1}.\frac{\mathrm{2}^{\mathrm{m}} −\mathrm{1}}{\mathrm{2}−\mathrm{1}}=\mathrm{2}^{\mathrm{m}} −\mathrm{1} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com