Find the sum of the following series : 5+(-41)+9+(-39). . .

Question : Find the sum of the following series : 5+(-41)+9+(-39)+13+(-37)+17+...+(-5)+81+(-3)

Doubt by Pushkar

Solution :

=5+(-41)+9+(-39)+13+(-37)+17+...+(-5)+81+(-3)

=[5+(-41)]+[9+(-39)]+[13+(-37)]+17+...+(-5)+[81+(-3)]

= (-36)+(-30)+(-24)+ . . .+(78)

Now

a₁=-36

a₂=-30

a₃=-24

d₁=a₂-a₁
= -30-(-36)
= -30+36
= 6

d₂=a₃-a₂
= -24-(-30)
= -24+30
= 6

d₁=d₂

Hence, this is an AP

Now
a = -36

d = 6

a_n = 78

a_n=a+(n-1)d
78=-36+(n-1)6

78+36=(n-1)6
114=(n-1)6

114/6=(n-1)
19=n-1

19+1=n

20=n

n=20

S_n=(n/2)[a+a_n]
S₂₀=(20/2)[-36+78]
S₂₀=10[42]
S₂₀=420

Hence, the sum of the entire series is 420.

Class 10