Infinite Sum
− 1 0 + 1 − 0 . 1 + 0 . 0 1 − 0 . 0 0 1 + 0 . 0 0 0 1 − … = ?
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4 solutions
Discussions for this problem are now closed
We can only apply if abs (r)<1.
Yes, this is true. However, ∣ ∣ ∣ ∣ − 1 0 1 ∣ ∣ ∣ ∣ ≤ 1 so we are safe to use it here :)
S = − 1 0 − ( − 1 + . 1 − . 0 1 + . . . ) S = − 1 0 − 1 0 S 1 0 S = − 1 0 0 − S 1 1 S = − 1 0 0 S = − 1 1 1 0 0
This is the sum of an infinite geometric sequence
The first term is
a = -10
The common ratio is
r = - 0.1
sum = a/(1 - r) = -(100/11)
Formula fir the Sum of an infinite geometric sequence:-
a+ar+ar^2+.........=a/(1-r). We are given the geometric sequence: -10+1-0.1+0.01-0.001+0.0001-...... =1+0.01+0.0001+...)-(10+0.1+0.001+....) Here,a=1 & r=0.01 and a' =10 & r' = 0.01 :-So we have, =[a/(1-r)] - [a'/(1-r')] =[1/1-0.01] - [10/1-0.01] =-100/11 .(required ans.)
Recall the formula for the sum of an infinite geometric sequence: a + a r + a r 2 + ⋯ = 1 − r a
We see that in this case, a = − 1 0 and r = − 1 0 1 so our sum is just S = 1 − ( − 1 0 1 ) − 1 0 = − 1 1 1 0 0