Question Number 56229 by Hassen_Timol last updated on 12/Mar/19
$$\mathrm{How}\:\mathrm{can}\:\mathrm{you}\:\mathrm{prove}\:\left(\mathrm{not}\:\mathrm{geometrically}\right) \\ $$$$\mathrm{the}\:\mathrm{following}? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\underset{{k}\:=\:\mathrm{0}} {\overset{{n}} {\sum}}{k}\:\:=\:\:\frac{\:{n}\:\left(\:{n}\:+\:\mathrm{1}\:\right)\:}{\mathrm{2}} \\ $$$$ \\ $$
Commented by Kunal12588 last updated on 12/Mar/19
$${Can}\:{I}\:{use}\:{Aritmetic}\:{Progression}\:{AP}\:{sir}. \\ $$
Commented by Hassen_Timol last updated on 12/Mar/19
$${yes}\:{Sir}\:{please} \\ $$
Commented by Kunal12588 last updated on 12/Mar/19
$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{k}=\mathrm{0}+\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+…+{n} \\ $$$${let}\:{us}\:{write}\:{the}\:{terms} \\ $$$$\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7},…,{n} \\ $$$${above}\:{sequence}\:{is}\:{in}\:{AP} \\ $$$${where}\:{a}=\mathrm{0},\:{d}=\mathrm{1} \\ $$$${S}_{{r}} =\frac{{r}}{\mathrm{2}}\left[\mathrm{2}{a}+\left({r}−\mathrm{1}\right){d}\right] \\ $$$$\therefore{S}_{{n}+\mathrm{1}} =\frac{\left({n}+\mathrm{1}\right)}{\mathrm{2}}\left[\mathrm{2}×\mathrm{0}+\left(\left({n}+\mathrm{1}\right)−\mathrm{1}\right)\left(\mathrm{1}\right)\right] \\ $$$$\Rightarrow{S}_{{n}+\mathrm{1}} =\frac{{n}+\mathrm{1}}{\mathrm{2}}\left(\mathrm{0}+{n}×\mathrm{1}\right)=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$
Commented by maxmathsup by imad last updated on 13/Mar/19
$${solution}\:{by}\:{polynome}\:{method}\:{let}\:{p}\left({x}\right)=\frac{{x}\left({x}−\mathrm{1}\right)}{\mathrm{2}}\:{we}\:{have}\:{for}\:{all}\:{x}\:{from}\:{R} \\ $$$${p}\left({x}+\mathrm{1}\right)−{p}\left({x}\right)=\frac{\left({x}+\mathrm{1}\right){x}}{\mathrm{2}}\:−\frac{{x}\left({x}−\mathrm{1}\right)}{\mathrm{2}}\:=\frac{{x}^{\mathrm{2}} +{x}\:−{x}^{\mathrm{2}} \:+{x}}{\mathrm{2}}\:={x}\:\Rightarrow \\ $$$$\forall\:{k}\in\left[\left[\mathrm{1},{n}\right]\right]\:{p}\left({k}+\mathrm{1}\right)−{p}\left({k}\right)\:={k}\:\Rightarrow\sum_{{k}=\mathrm{1}} ^{{n}} \left\{{p}\left({k}+\mathrm{1}\right)−{p}\left({k}\right)\right\}=\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}\:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} {k}\:={p}\left(\mathrm{2}\right)−{p}\left(\mathrm{1}\right)+{p}\left(\mathrm{3}\right)−{p}\left(\mathrm{2}\right)+….+{p}\left({n}+\mathrm{1}\right)−{p}\left({n}\right)={p}\left({n}+\mathrm{1}\right)−{p}\left(\mathrm{1}\right) \\ $$$$=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:−\mathrm{0}\:\Rightarrow\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}\:=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:. \\ $$$$ \\ $$
Answered by Prithwish sen last updated on 12/Mar/19
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{k}\:=\:\mathrm{1}+\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:+\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:+…………+\left(\mathrm{n}−\mathrm{2}\right)+\left(\mathrm{n}−\mathrm{1}\right)+\mathrm{n} \\ $$$$\:\mathrm{again}\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{k}\:=\:\mathrm{n}+\left(\mathrm{n}−\mathrm{1}\right)+\left(\mathrm{n}−\mathrm{2}\right)\:+………….+\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:+\:\:\:\mathrm{2}\:\:\:\:\:\:\:+\mathrm{1}\:\:\:\:\:\: \\ $$$$\mathrm{adding}\:\mathrm{2}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}=\:\left(\mathrm{n}+\mathrm{1}\right)+\left(\mathrm{n}+\mathrm{1}\right)+….\mathrm{upto}\:\mathrm{n}\:\mathrm{times} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\therefore\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}=\frac{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)}{\mathrm{2}}\:\:\:\mathrm{proved} \\ $$
Commented by Hassen_Timol last updated on 12/Mar/19
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Sir},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$
Answered by Kunal12588 last updated on 12/Mar/19
$$ \\ $$$${S}_{{n}+\mathrm{1}} =\underset{{k}\:=\:\mathrm{0}} {\overset{{n}} {\sum}}{k}=\mathrm{0}+\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+…+{n}\:\:\:\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$${There}\:{are}\:\left({n}+\mathrm{1}\right)\:{terms}\:{in}\:{above}\:{series}. \\ $$$${Reversing}\:\:{S}_{{n}} \:\because\:{addition}\:{follows}\:{commutive} \\ $$$${property}\:{it}\:{does}\:{not}\:{effect}\:{the}\:{S}_{{n}} \\ $$$${S}_{{n}+\mathrm{1}} ={n}+\left({n}−\mathrm{1}\right)+\left({n}−\mathrm{2}\right)+…+\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$${adding}\:{corresponding}\:{terms}\:{of}\:\left(\mathrm{1}\right)\:{and}\:\left(\mathrm{2}\right) \\ $$$$\mathrm{2}{S}_{{n}+\mathrm{1}} =\left(\mathrm{0}+{n}\right)+\left[\mathrm{1}+\left({n}−\mathrm{1}\right)\right]+\left[\mathrm{2}+\left({n}−\mathrm{2}\right)\right]+..+\left({n}+\mathrm{0}\right) \\ $$$${there}\:{are}\:\left({n}+\mathrm{1}\right)\:{brackets} \\ $$$${as}\:{you}\:{can}\:{see}\:{every}\:{bracket}\:{will}\:{give}\:{us}\:'{n}' \\ $$$$\mathrm{2}{S}_{{n}+\mathrm{1}} ={n}+{n}+{n}+{n}+…\left({n}+\mathrm{1}\right)\:{terms} \\ $$$$\mathrm{2}{S}_{{n}+\mathrm{1}} ={n}\left({n}+\mathrm{1}\right) \\ $$$${S}_{{n}+\mathrm{1}} =\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$
Commented by Hassen_Timol last updated on 12/Mar/19
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Sir},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$
Commented by Prithwish sen last updated on 12/Mar/19
$$\mathrm{this}\:\mathrm{is}\:\mathrm{correct} \\ $$
Commented by Kunal12588 last updated on 12/Mar/19
$${May}\:{God}\:{bless}\:{you}\:{too}. \\ $$$${Welcome}. \\ $$