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Bunuel
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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? 11 Jun 2019, 01:53
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D
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77% (01:16) correct 23% (01:21) wrong based on 273 sessions

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If \(S_n = 4^n + 5^{(n + 1)} + 3\) what is the unit digit of \(s_{100}\)?

A. 1
B. 2
C. 3
D. 4
E. 6
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D
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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? Updated on: 11 Jun 2019, 02:20
Originally posted by Lampard42 on 11 Jun 2019, 01:58.
Last edited by Lampard42 on 11 Jun 2019, 02:20, edited 1 time in total.
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Bunuel
If \(S_n = 4^n + 5^{(n + 1)} + 3\) what is the unit digit of \(s_{100}\)?

A. 1
B. 2
C. 3
D. 4
E. 6
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Using the unit digits are cyclic i.e. units digit of 4 repeats itself every 2 powers and 5 unit digit remains 5 for any power. So 6+5+3=14 IMO D
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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? 11 Jun 2019, 02:04
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4 ^ 100 = (4^2) ^50 --- last digit 6
5 ^ anything ---------- last digit 5

Last digits of the given expression 6 + 5+ 3 = 4

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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? 11 Jun 2019, 02:10
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S100 = 4^100 + 5^101 + 3

Now, 4^100 = (2^2)^100 = 2^200.

Unit digits:

200 = 4n therefore, unit digit of 2^200 = 6

Unit digit for 5^101 = 5

Unit digit for 3^1 = 3

Add the individual unit digit: 6+5+3 = 14

Hence, unit digit of S100 is 4

Answer: D
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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? 12 Jun 2019, 03:05
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Solution

Given:
• \(S_n = 4^n + 5^{n + 1} + 3\)

To Find:
• The unit digit of \(S_{100}\)

Approach & Working Out:
    • The unit digit of \(S_{100}\) = The unit digit of \(4^{100} + 5^{101} + 3\) = unit digit of 6 + 5 + 3 = 4

Hence, the correct answer is Option D
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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? 17 Jun 2019, 17:33
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Bunuel
If \(S_n = 4^n + 5^{(n + 1)} + 3\) what is the unit digit of \(s_{100}\)?

A. 1
B. 2
C. 3
D. 4
E. 6
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We may recall that 4 raised to an even power ends in 6, 4 raised to an odd power ends in 4, and 5 raised to any whole number power, ends in 5, Thus:

If we add the units digits of 4^100, 5^101, and 3, the sum will be 6 + 5 + 3 = 14. Therefore, 4^100 + 5^101 + 3 has a units digit of 4.

Answer: D
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If Sn = 4^n + 5^(n + 1) + 3 what is the unit digit of s_{100}? 19 Dec 2020, 11:42
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