Question

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Question 13 Jason has an investment of 1,000 that accumulates at a force of interest of 8, = 0.02 +0.004. Let A be the accumulated value at time n, and B be the amount of interest earned during the sixth year. You are given that А + 10B = 1,346.13 Determine n.

Answer 1

Answer :-

Given

Investment = 1000
Interest = 0.02 + 0.004 t
Interest in 6th year
Interest in 1st year = 0.02 + 0.004 (1) = 0.024 = 2.4 %
Interest in 2nd year = 0.02 + 0.004 (2) = 0.02 + 0.008 = 0.028 = 2.8%
Interest in 3rd year = 0.02 + 0.004 (3) = 0.02 + 0.012 = 0.032 = 3.2 %
Interest in 4th year = 0.02 + 0.004 (4) = 0.02 + 0.016 = 0.036 = 3.6 %
Interest in 5th year = 0.02 + 0.004 (5) = 0.02 + 0.02 = 0.04 = 4 %
Interest in 6th year = 0.02 + 0.004 (6) = 0.02 + 0.024 = 0.044 = 4.4 %
Interest in 7th year = 0.02 + 0.004 (7) = 0.02 + 0.028 = 0.048 = 4.8 %
Interest in 8th year = 0.02 + 0.004 (8) = 0.02 + 0.032 = 0.052 = 5.2 %
Interest in 9th year = 0.02 + 0.004 (9) = 0.02 + 0.036 = 0.056= 5.6 %
Interest in 10th year = 0.02 + 0.004 (10) = 0.02 + 0.04 = 0.06 = 6 %
​​​​​​​Interest in 11th year = 0.02 + 0.004 (11) = 0.02 + 0.044 = 0.064 = 6.4 %
​​​​​​​Interest in 12th year = 0.02 + 0.004 (12) = 0.02 + 0.048 = 0.068 = 6.8%
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Amount at the end of 1st year = 1000 x (1.024) = 1024
Amount at the end of 2nd year = 1024 x ( 1.028) = 1052.67
Amount at the end of 3rd year = 1052.67 x (1.032) = 1086.35
Amount at the end of 4th year = 1086.35 x (1.036) = 1125.46
Amount at the end of 5th year = 1125.46 x (1.04) = 1170.48
Amount at the end of 6th year = 1170.48 x (1.044) = 1221.98

B = sixth year interest = 1221.98 - 1170.48 = 51.5
10 B = 10 x B = 10 x 51.5 = 515

A / 2 + 10 B = 1346.13
A / 2 + 515 = 1346.13
A / 2 = 1346.13 - 515
A / 2 = 831.13
A = 831.13 x 2 = 1662.26

Amount at the end of 7th year = 1221.98 x (1.048) = 1280.64
Amount at the end of 8th year = 1280 x (1.052) = 1347.23
Amount at the end of 9th year = 1347.23 x (1.056) = 1422.67
Amount at the end of 10th year = 1422.67 x (1.06) = 1508.03
Amount at the end of 11th year = 1508.03 x (1.064) = 1604.54
Amount at the end of 12th year = 1604.54 x (1.068) = 1713.64 [ This value exceeds A= 1662.26]

Therefore n = 12
The correct option is c.