Sqrt between ints

Which two consecutive integers does the square root lie in? *

√45
√23
√71
√7
√86

Final Answer:

a₄₅ = 6
b₄₅ = 7
a₂₃ = 4
b₂₃ = 5
a₇₁ = 8
b₇₁ = 9
a₇ = 2
b₇ = 3
a₈₆ = 9
b₈₆ = 10

a_{45} = ⌊ 45 ⌋ = 6 a_{45} = s_{45} = a_{45}^{2} = 6^{2} = 36 = 6

b_{45} = a_{45} + 1 = 6 + 1 = 7 S_{45} = b_{45}^{2} = 7^{2} = 49 b_{45} = 6 < 45 = 6.7082039324994 < 7 = 7

a_{23} = ⌊ 23 ⌋ = 4 a_{23} = s_{23} = a_{23}^{2} = 4^{2} = 16 = 4

b_{23} = a_{23} + 1 = 4 + 1 = 5 S_{23} = b_{23}^{2} = 5^{2} = 25 b_{23} = 4 < 23 = 4.7958315233127 < 5 = 5

a_{71} = ⌊ 71 ⌋ = 8 a_{71} = s_{71} = a_{71}^{2} = 8^{2} = 64 = 8

b_{71} = a_{71} + 1 = 8 + 1 = 9 S_{71} = b_{71}^{2} = 9^{2} = 81 b_{71} = 8 < 71 = 8.4261497731764 < 9 = 9

a_{7} = ⌊ 7 ⌋ = 2 a_{7} = s_{7} = a_{7}^{2} = 2^{2} = 4 = 2

b_{7} = a_{7} + 1 = 2 + 1 = 3 S_{7} = b_{7}^{2} = 3^{2} = 9 b_{7} = 2 < 7 = 2.6457513110646 < 3 = 3

a_{86} = ⌊ 86 ⌋ = 9 a_{86} = s_{86} = a_{86}^{2} = 9^{2} = 81 = 9

b_{86} = a_{86} + 1 = 9 + 1 = 10 S_{86} = b_{86}^{2} = 1 0^{2} = 100 b_{86} = 9 < 86 = 9.2736184954957 < 10 = 10

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