ধারা
যোগফল নির্ণয় করোঃ 1+(1+15)+(1+15+152)+(1+15+152+153)+….. 1 + \left ( 1 + \frac{1}{5} \right ) + \left ( 1 + \frac{1}{5} + \frac{1}{5^{2}} \right ) + \left ( 1 + \frac{1}{5} + \frac{1}{5^{2}} + \frac{1}{5^{3}} \right ) + \ldots . . 1+(1+51)+(1+51+521)+(1+51+521+531)+….. +n তম পদ পর্যন্ত
Un=1+15+152+………+15n−1=1(1−15n)1−15=54(1−15n) \mathrm{U}_{\mathrm{n}}=1+\frac{1}{5}+\frac{1}{5^{2}}+\ldots \ldots \ldots+\frac{1}{5^{\mathrm{n}-1}}=\frac{1\left(1-\frac{1}{5^{\mathrm{n}}}\right)}{1-\frac{1}{5}}=\frac{5}{4}\left(1-\frac{1}{5^{\mathrm{n}}}\right) Un=1+51+521+………+5n−11=1−511(1−5n1)=45(1−5n1)
Sn=54(∑1−∑15n)=54[n−(15+152+153+………+15n)]=54[n−15(1+15+………+15n−1)]=54[n−15{54(1−15n)}] \begin{array}{l} S_{n}=\frac{5}{4}\left(\sum 1-\sum \frac{1}{5^{\mathrm{n}}}\right)=\frac{5}{4}\left[\mathrm{n}-\left(\frac{1}{5}+\frac{1}{5^{2}}+\frac{1}{5^{3}}+\ldots \ldots \ldots+\frac{1}{5^{\mathrm{n}}}\right)\right] \\ =\frac{5}{4}\left[\mathrm{n}-\frac{1}{5}\left(1+\frac{1}{5}+\ldots \ldots \ldots+\frac{1}{5^{\mathrm{n}-1}}\right)\right]=\frac{5}{4}\left[\mathrm{n}-\frac{1}{5}\left\{\frac{5}{4}\left(1-\frac{1}{5^{\mathrm{n}}}\right)\right\}\right] \end{array} Sn=45(∑1−∑5n1)=45[n−(51+521+531+………+5n1)]=45[n−51(1+51+………+5n−11)]=45[n−51{45(1−5n1)}]
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