(i) 7 + 10 ½ + 14 + . . . + 84
(ii) 34 + 32 + 30 + . . . + 10
(iii) –5 + (–8) + (–11) + . . . + (–230)
Solution:
(i) 7 + 10 ½ + 14 + . . . + 84
First term, a = 7
nth term, an = 84
Common difference, d = a2 – a1 = 10 ½ – 7 = (21/2) – 7 = 7/2
Let 84 be the nth term of this AP, then as per the nth term formula,
an = a + (n – 1)d
84 = 7 + (n – 1) × 7/2
77 = (n – 1) × 7/2
22 = n − 1
n = 23
We know that, sum of n term is;
Sn = n/2 (a + l) , where l = 84
Sn = 23/2 (7 + 84)
Sn = (23 × 91/2) = 2093/2
Sn = 1046 ½
(ii) 34 + 32 + 30 + ……….. + 10
For this A.P.,
first term, a = 34
common difference, d = a2− a1 = 32 − 34 = −2
nth term, an= 10
Let 10 be the nth term of this AP, therefore,
an = a + (n − 1)d
10 = 34 + (n − 1)(−2)
−24 = (n − 1)(−2)
12 = n −1
n = 13
We know that the sum of n terms is;
Sn = n/2 (a + l), where l = 10
= 13/2 (34 + 10)
= (13 × 44/2) = 13 × 22
= 286
(iii) (−5) + (−8) + (−11) + ………… + (−230)
For this A.P.,
First term, a = −5
nth term, an= −230
Common difference, d = a2 − a1 = (−8) − (−5)
⇒d = −8 + 5 = −3
Let −230 be the nth term of this AP, and by the nth term formula we know,
an = a + (n − 1) d
−230 = −5 + (n − 1)(−3)
−225 = (n − 1)(−3)
(n − 1) = 75
n = 76
And, Sum of n term,
Sn = n /2 ( a + l )
= 76/2 [(-5) + (-230)]
= 38(-235)
= -8930