Question

Calculate the sum of the sequences given below:
$(a)\;7 + 10\dfrac{1}{2} + 14 + ... + 84$
$(b)\;34 + 32 + 30 + ... + 10$
$(c)\; - 5 + ( - 8) + ( - 11) + ... + ( - 230)$

Answer 1

Each of these subdivisions can be solved in the same manner. We can see that each of these questions show a similar pattern. The pattern the series are showing is that they are having a common difference between consecutive terms. Common difference means that the series shows an arithmetic progression. Remember that the formula for sum of $n$terms of a series is:
$\Rightarrow Sn = \dfrac{n}{2}[2 \times a + (n - 1) \times d]$, here $a =$first term in the sequence, $d =$common difference between terms, $n =$number of terms in the sequence

Complete step-by-step solution:
$(a)$ First let us check what kind of pattern the series follows:
First three terms
$a_1 = 7,\;a_2 = 10\dfrac{1}{2},\;a_3 = 14$
Checking the difference between consecutive terms:
$\Rightarrow a_2 - a_1 = \dfrac{{21}}{2} - 7 = 3.5 \\\ \Rightarrow a_3 - a_2 = 14 - \dfrac{{21}}{2} = 3.5 \\\$
So there is a common difference, let it be $d = 3.5$
To find number of terms $n$, we can use the formula;
$an = a + (n - 1) \times d$
$\Rightarrow 84 = 7 + (n - 1) \times 3.5$
$\Rightarrow n = 23$
Now clearly sum of $\Rightarrow n = 23$ terms will be;
$\Rightarrow Sn = \dfrac{{23}}{2}[2 \times 7 + (23 - 1) \times 3.5]$
$\Rightarrow S23 = 1046\dfrac{1}{2}$
So the sum of this sequence is: $1046\dfrac{1}{2}$

$(b)$ First let us check what kind of pattern the series follows:
First three terms
$a_1 = 34,\;a_2 = 32,\;a_3 = 30$
Checking the difference between consecutive terms:
$\Rightarrow a_2 - a_1 = 34 - 32 = - 2 \\\ \Rightarrow a_3 - a_2 = 30 - 32 = - 2 \\\$
So there is a common difference, let it be $d = - 2$
To find number of terms $n$, we can use the formula;
$an = a + (n - 1) \times d$
$\Rightarrow 10 = 34 + (n - 1) \times - 2$
$\Rightarrow n = 13$
Now clearly sum of $\Rightarrow n = 13$ terms will be;
$\Rightarrow Sn = \dfrac{{13}}{2}[2 \times 34 + (13 - 1) \times - 2]$
$\Rightarrow S13 = 286$
So the sum of this sequence is: $286$

$(c)$ First let us check what kind of pattern the series follows:
First three terms
$a_1 = - 5,\;a_2 = - 8,\;a_3 = - 11$
Checking the difference between consecutive terms:
$\Rightarrow a_2 - a_1 = - 8 + 5 = - 3 \\\ \Rightarrow a_3 - a_2 = - 11 + 8 = - 3 \\\$
So there is a common difference, let it be $d = - 3$
To find number of terms $n$, we can use the formula;
$an = a + (n - 1) \times d$
$\Rightarrow - 230 = - 5 + (n - 1) \times - 3$
$\Rightarrow n = 76$
Now clearly sum of $\Rightarrow n = 76$ terms will be;
$\Rightarrow Sn = \dfrac{{76}}{2}[2 \times - 5 + (76 - 1) \times - 3]$
$\Rightarrow S23 = - 8930$
So the sum of this sequence is: $- 8930$

Note: The division of the numerator by the denominator can be a cumbersome task as it involves thorough knowledge of algebraic rules and long division method. Care should be taken while doing the same and proceeding with the process to convert a fraction into decimal. The decimal expansion of a rational number in $\dfrac{p}{q}$ form can be terminating or recurring in nature.