2¹+2²+2³+2⁴+2⁵..+2⁹⁹
Which is unit place number?
Answers
Concept : - UNITS DIGIT USING CYCLICITY
Every number after multiplying itself for a given number of times the same number gets repeated in the units place which we call cyclicity.
Numbers cyclicity
0 , 1 , 5 , 6 1
2 , 3 , 7 , 8 4
4 , 9 2
Solution : -
Here we have powers of digit 2 so units digits gets repeated with cyclicity 4 with power cycles as 2 , 4 , 8 , 6 .
units digit of 2¹ = 2
units digit of 2² = 4
units digit of 2³ = 8
units digit of 2⁴ = 6
units digit of 2⁵ = 2....
Total no. of terms = 99
no. of complete power cycles = 99/4 = 24
last three will have units digits of 2 , 4 , 8
so sum of units places of all the terms ,
⇒ S = 24 x ( 2 + 4 + 8 + 6 ) + 2 +4 + 8
⇒ S = 24 x ( 20) + 14
⇒S = 494
Hence the units place is 4
∴ units digit of the sum of the series 2¹+2²+2³+2⁴+2⁵_____+2⁹⁹ = 4
Answer:
What is the digit in the unit place of 2⁹⁹. ?