Question

Find the sums given below:
(i) 7+$\frac{21}{2}$ +14+...+84
(ii) 34+32+30+...+10
(iii) -5 + (-8) + (-11) + .... + (-230)

Answer 1

(i) 7+$\frac{21}{2}$ +14+...+84

First term = a = 7

Common difference = d = $\frac{21}{2}$ −7= $\frac{(21-14)}{2}$ =$\frac{7}{2}$ =3.5

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 $n^{th}$ term of arithmetic progression, we can say that

(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=$\frac{80.5}{3.5}$ =23

Therefore, there are 23 terms in the given AP. It means n = 23.

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

$S_{23}$=$\frac{23}{2}$ (7+84)

$\Rightarrow$$S_{23}$=$\frac{23}{2}$ $\times$ 91=1046.5

(ii) 34+32+30+...+10

First term = a = 34

Common difference = d = 32 - 34 = -2

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 $n^{th}$ term of arithmetic progression, we can say that

(34+ (n−1) (−2)) =10

$\Rightarrow$34−2n+2=10

$\Rightarrow$−2n=−26

$\Rightarrow$n=$\frac{26}{2}$ =13

Therefore, there are 13 terms in the given AP. It means n = 13.

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

$S_{13}$=$\frac{13}{2}$ (34+10) =$\frac{13}{2}$ ×44=286

(iii) −5+ (−8) + (−11) +....+ (−230)

First term = a = -5

Common difference = d = -8 - (-5) = -8 + 5 = -3

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 $n^{th}$ term of arithmetic progression, we can say that

(−5+ (n−1) (−3))=−230

$\Rightarrow$−5−3n+3=−230

$\Rightarrow$−3n=−228

$\Rightarrow$n=$\frac{228}{3}$ =76

Therefore, there are 76 terms in the given AP. It means n = 76.

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

$S_{76}$=$\frac{76}{2}$ (−5−230) =$\frac{76}{2}$ $\times$ (−235) =−8930