Question

Find the sum of each arithmetic series
(i) $38+35+32+.....+2$
(ii) $6+5\frac{1}{4}+4\frac{1}{2}+....25terms$

Answer 1

(i) The given arithmetic series is $38+35+32+....+2$ where the first term is $a_1=38$, second term is $a_2=35$ and the $n$th term is $T_n=2$.

We find the common difference $d$ by subtracting the first term from the second term as shown below:

$d=a_2-a_1=35-38=-3$

We know that the general term of an arithmetic progression with first term $a$ and common difference $d$ is $T_n=a+(n-1)d$, therefore, with $a=38,d=-3$ and $T_n=2$, we have

$T_n=a+(n-1)d\Rightarrow 2=38+(n-1)(-3)\Rightarrow 2=38-3n+3\Rightarrow 2=41-3n\Rightarrow 3n=41-2\Rightarrow 3n=39\Rightarrow n=\frac{39}{3}\Rightarrow n=13$

We also know that the sum of an arithmetic series with first term $a$ and common difference $d$ is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Now to find the sum of series, substitute $n=13,a=38$ and $d=-3$ in $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$ as follows:

$S_{13}=\frac{13}{2}\left[\right(2\times 38)+(13-1\left)\right(-3\left)\right]=\frac{13}{2}[76+(12\times -3\left)\right]=\frac{13}{2}(76-36)=\frac{13}{2}\times 40=13\times 20=260$

Hence, the sum of the given arithmetic series is $260$.

(ii) The given arithmetic series is $6+5\frac{1}{4}+4\frac{1}{2}+........$ where the first term is $a_1=6$, second term is $a_2=5\frac{1}{4}$ and the number of terms $n=25$.

We find the common difference $d$ by subtracting the first term from the second term as shown below:

$d=a_2-a_1=5\frac{1}{4}-6=\frac{21}{4}-6=\frac{21-24}{4}=-\frac{3}{4}$

We know that the sum of an arithmetic series with first term $a$ and common difference $d$ is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Now to find the sum of series, substitute $n=25,a=6$ and $d=-\frac{3}{4}$ in $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$ as follows:

$S_{25}=\frac{25}{2}\left[\right(2\times 6)+(25-1\left)(-\frac{3}{4})\right]=\frac{25}{2}[12+(24\times -\frac{3}{4})]=\frac{25}{2}[12-18]=\frac{25}{2}\times -6=25\times -3=-75$

Hence, the sum of the given arithmetic series is $-75$.