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Zorluk: ZorEstimation, Rounding, and Sequences

An index value I0=1000I_0 = 1{}000 increases over a 6-month period. For each month nn from 1 to 6, the index value InI_n is calculated by increasing the previous month's value In−1I_{n-1} by 5%5\% and then rounding the result to the nearest integer (with half-integers rounded up). Let U6=1000×(1.05)6U_6 = 1{}000 \times (1.05)^6 represent the exact unrounded compounded value at month 6.

Which of the following statements must be true? Select all such statements.

  1. I2=1103I_2 = 1{}103Cevap
  2. B
    The exact unrounded value U6=1000×(1.05)6U_6 = 1{}000 \times (1.05)^6 is strictly greater than the rounded index value I6I_6.
  3. The sequence of monthly increments In−In−1I_n - I_{n-1} for n=1,2,…,6n = 1, 2, \dots, 6 is strictly increasing.Cevap
  4. D
    The sum of the first two monthly fractional growth rates, I1−I0I0+I2−I1I1\frac{I_1 - I_0}{I_0} + \frac{I_2 - I_1}{I_1}, is equal to I2−I0I0+I1\frac{I_2 - I_0}{I_0 + I_1}.
  5. E
    The exponential growth factor (1+0.05)6(1 + 0.05)^6 is equivalent to 16+(0.05)61^6 + (0.05)^6.

Cevap

The statement specifying that I2=1103I_2 = 1{}103 and the statement asserting that the sequence of monthly increments In−In−1I_n - I_{n-1} is strictly increasing are both correct.
The statement giving I2=1103I_2 = 1{}103 is correct because 1050×1.05=1102.51{}050 \times 1.05 = 1{}102.5, which rounds up to 11031{}103. The statement regarding the sequence of monthly increments is correct because the increments 50,53,55,58,61,6450, 53, 55, 58, 61, 64 strictly increase.

Adım Adım Çözüm

1
Calculate each term of the sequence InI_n by applying a 5% increase and rounding to the nearest integer.
I0=1000I_0 = 1{}000; I1=round(1000×1.05)=1050I_1 = \text{round}(1{}000 \times 1.05) = 1{}050; I2=round(1050×1.05)=round(1102.5)=1103I_2 = \text{round}(1{}050 \times 1.05) = \text{round}(1{}102.5) = 1{}103; I3=round(1103×1.05)=round(1158.15)=1158I_3 = \text{round}(1{}103 \times 1.05) = \text{round}(1{}158.15) = 1{}158; I4=round(1158×1.05)=round(1215.9)=1216I_4 = \text{round}(1{}158 \times 1.05) = \text{round}(1{}215.9) = 1{}216; I5=round(1216×1.05)=round(1276.8)=1277I_5 = \text{round}(1{}216 \times 1.05) = \text{round}(1{}276.8) = 1{}277; I6=round(1277×1.05)=round(1340.85)=1341I_6 = \text{round}(1{}277 \times 1.05) = \text{round}(1{}340.85) = 1{}341.
This establishes the exact sequence of rounded monthly values.
2
Evaluate the statement that I2=1103I_2 = 1{}103.
From Step 1, I2=1103I_2 = 1{}103.
This directly confirms the validity of the first statement.
3
Compute the sequence of monthly increments In−In−1I_n - I_{n-1} for n=1,2,…,6n = 1, 2, \dots, 6.
Increments: I1−I0=50I_1 - I_0 = 50, I2−I1=53I_2 - I_1 = 53, I3−I2=55I_3 - I_2 = 55, I4−I3=58I_4 - I_3 = 58, I5−I4=61I_5 - I_4 = 61, I6−I5=64I_6 - I_5 = 64.
Since 50<53<55<58<61<6450 < 53 < 55 < 58 < 61 < 64, the sequence of increments is strictly increasing.
4
Compare I6I_6 with U6=1000×(1.05)6U_6 = 1{}000 \times (1.05)^6.
U6=1000×1.3400956...≈1340.10U_6 = 1{}000 \times 1.3400956... \approx 1{}340.10. Since I6=1341I_6 = 1{}341, I6>U6I_6 > U_6.
The statement claiming U6>I6U_6 > I_6 is false.

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Estimation, Rounding, and Sequences

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