Find the sums given below:
(i) \(7 + {10}{\dfrac{1}{2}} + 14 + .......+ 84\)
(ii) 34 + 32 + 30 + .....… + 10
(iii) –5 + (–8) + (–11) + …..... + (–230)

(i) \(7 + {10}{\dfrac{1}{2}} + 14 +..........+ 84\)

Here First term = a = 7,
Common difference = d = \({{21} \over {2}} - 7 = {{21 - 15} \over {2}} = {{7} \over {2}} = 3.5\)

And Last term = l = 84

We do not know how many terms are there in the given AP.

So, we need to find n first.

Using formula \(a_n = a + (n - 1)d\) , to find nth term of arithmetic progression,

\(\Rightarrow\) [7 + (n - 1) (3.5)] = 84
\(\Rightarrow \) 7 + (3.5) n - 3.5 = 84
\(\Rightarrow \) 3.5n = 84 + 3.5 – 7
\(\Rightarrow \) 3.5n = 80.5
\(\Rightarrow \) n = 23

Therefore, there are 23 terms in the given AP.

It means n = 23.

Applying formula \(S_n = {{n} \over {2}} (a + l)\), to find sum of n terms of AP,

\(\Rightarrow S_n = {{23} \over {2}} (7 + 84)\)
\(\Rightarrow S_{23} = {{23} \over {2}} (91) = 1046.5\)


(ii) 34 + 32 + 30 + … + 10
Here First term = a = 34,
Common difference = d = 32 – 34 = –2
And Last term = l = 10

We do not know how many terms are there in the given AP.

So, we need to find n first.

Using formula \(a_n = a + (n - 1)d\), to find nth term of arithmetic progression,

\(\Rightarrow\) [34 + (n - 1) (-2)] = 10
\(\Rightarrow \) 34 – 2n + 2 = 10
\(\Rightarrow \) 2n = 26
\(\Rightarrow \) n = 13

Therefore, there are 13 terms in the given AP.

It means n = 13.

Applying formula \(S_n = {{n} \over {2}} (a + l)\), to find sum of n terms of AP,

\( S_{13} = {{13} \over {2}} (34 + 10) \)
\(= {{13} \over {2}} (44) = 286\)


(iii) -5 + (-8) + (-11) + … + (-230)

Here First term = a = –5,
Common difference = d
= –8 – (–5) = –8 + 5 = –3

And Last term = l = -230

We do not know how many terms are there in the given AP.

So, we need to find n first.

Using formula \(a_n = a + (n - 1)d\), to find nth term of arithmetic progression,

\(\Rightarrow\) [-5 + (n - 1) (-3)] = -230
\(\Rightarrow \) -5 - 3n + 3 = -230
\(\Rightarrow\) -3n = -228
\(\Rightarrow \) n = 76

Therefore, there are 76 terms in the given AP.

It means n = 76.

Applying formula \(S_n = {{n} \over {2}} (a + l)\), to find sum of n terms of AP,
\(S_{76} = {{76} \over {2}} (-5 - 230) \)
\( = 38 (-235) = -8930\)