A sequence of 40 numbers a_1, a_2, \dots, a_{40} is defined by the formula a_k = 15 + \fra… — GRE General Test

A sequence of 40 numbers $a_1, a_2, \dots, a_{40}$ is defined by the formula $a_k = 15 + \frac{3k}{8}$ for each integer $k$ from 1 to 40. Each term $a_k$ is rounded to the nearest integer to form a new sequence $b_1, b_2, \dots, b_{40}$ . (Note: numbers ending in $.5$ are rounded up to the next integer.) What is the value of $\sum_{k=1}^{40} b_k - \sum_{k=1}^{40} a_k$ ?

$2.5$ Answer
B
$0.0$
C
$5.0$
D
$-2.5$
E
$1.0$

Answer

The sum of the rounded sequence exceeds the sum of the exact sequence by

2.5

Evaluating

a_k = 15 + \frac{3k}{8}

for

k = 1, 2, \dots, 8

yields fractional parts of

0.375, 0.75, 0.125, 0.5, 0.875, 0.25, 0.625,

and

0.0

. The individual rounding errors

(b_k - a_k)

for these terms are

-0.375, +0.25, -0.125, +0.5, +0.125, -0.25, +0.375,

and

0.0

. Summing these errors yields

+0.5

per 8-term period. For 40 terms (5 full periods), the total error sum is

5 \times 0.5 = 2.5

Step-by-Step Solution

Analyze the fractional part of

a_k = 15 + \frac{3k}{8}

over one complete 8-term period.

The fractional parts for

k = 1, 2, \dots, 8

are

0.375, 0.75, 0.125, 0.5, 0.875, 0.25, 0.625, 0.0

respectively.

Since

\frac{3k}{8}

repeats its fractional pattern modulo 8, examining one period reveals the periodic rounding behavior.

Calculate the rounding error

(b_k - a_k)

for each term in the 8-term cycle.

The differences

(b_k - a_k)

for

k=1

8

are:

-0.375, +0.25, -0.125, +0.5, +0.125, -0.25, +0.375, 0.0

Rounding to the nearest integer shifts each number by its distance to that integer. Half-integers like

0.5

round up, yielding a

+0.5

difference.

Sum the rounding errors over one 8-term cycle.

(-0.375 + 0.375) + (0.25 - 0.25) + (-0.125 + 0.125) + 0.5 + 0.0 = +0.5

Symmetric fractional pairs cancel out, leaving only the

+0.5

error from the half-integer term.

Multiply the single-cycle error sum by the number of full cycles in 40 terms.

Since

40 = 5 \times 8

, total difference

= 5 \times 0.5 = 2.5

The 40 terms consist of 5 identical 8-term periodic cycles.

Key Concept

Periodic error analysis in sequence rounding and summation

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