Find the sum of AP 7+10 1/2+14...+84
Answers
Step-by-step explanation:
AP ( Arithmetic progression).
A list of numbers a1, a2, a3 ………….. an is called an arithmetic progression ,
Is there exists a constant number ‘d’
a2= a1+d
a3= a2+d
a4= a3+d……..
an= an-1+d ………
Each of the numbers in the list is called a term .
An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed number is called the common difference( d ) of the AP.
General form of an AP.:
a, a+d, a+2d, a+3d…….
Here a is the first term and d is common difference.
Sum of n terms of an AP
The sum of first n terms of an AP with first term 'a' and common difference 'd' is given by
Sn = n /2 [ 2a + ( n - 1) d] or
Sn=n /2 [ a + l ] (l = last term)
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Solution:
(i)Given:
a = 7
l = an= 84
d = a2 − a1 = – 7 = 21/2 – 7 = 7/2
d= 7/2
Let 84 be the nth term of this A.P.
l = a (n – 1)d
84 = 7 + (n – 1) × 7/2
77 = (n – 1) × 7/2
22 = n − 1
n = 23
We know that,
Sn = n/2 (a + l)
Sn = 23/2 (7 + 84)
= (23×91/2) = 2093/2
Sn= 1046 1/2
(ii) 34 + 32 + 30 + ……….. + 10
Given:
a = 34
d = a2 − a1 = 32 − 34 = −2d= -2l = an= 10
Let 10 be the nth term of this A.P.
l = a + (n − 1) d
10 = 34 + (n − 1) (−2)
−24 = (n − 1) (−2)
12 = n − 1
n = 13
Sn = n/2 (a + l)
= 13/2 (34 + 10)
= (13×44/2) = 13 × 22
Sn= 286
(iii) (−5) + (−8) + (−11) + ………… + (−230)
Given:
a = −5
l = −230
d = a2 − a1 = (−8) − (−5)
= − 8 + 5 = −3
d= -3
Let −230 be the nth term of this A.P.
l = a + (n − 1)d
−230 = − 5 + (n − 1) (−3)
−225 = (n − 1) (−3)
(n − 1) = 75
n = 76
Sn = n/2 (a + l)
= 76/2 [(-5) + (-230)] = 38×(-235)
Sn= -8930
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Hope this will help you