GMAT Official Guide Quantitative Review 2019_ Book

GMAT® Official Guide 2019 Quantitative Review

Arithmetic Prc,perties of n 1mber

The factors of 28 are 1, 2, 4, 7, 14, and 28.

Therefore, the sum of the reciprocals of the

1 1 1 1 1 1

factors of 28 is - + - + - + - + - + - =

1 2 4 7 14 28

28 14 7 4 2 1

-+-+-+-+-+-=

28 28 28 28 28 28

28 + 14 + 7 + 4 + 2 + 1 = 56 = 2.

28 28

The correct answer is C.

PS] 1430

137. The infinite sequence a 1 , a 2 , ... , an, ... is such that
     a 1 = 2, a 2 = -3, a3 = 5, a 4 = -1, and an= an_ 4 for
     n > 4. What is the sum of the first 97 terms of the
     sequence?

(Al 72

(Bl 74

(Cl 75

(Dl 78

(El 80

Arithmetic Sequ c - a d series

Because an = an_ 4 for n > 4, it follows that the

terms of the sequence repeat in groups of 4 terms:

Values for n Values for an

1,2,3,4 2,-3,5,-1

5,6, 7,8 2,-3,5,-1

9,10,11, 12 2,-3,5,-1

13,14,15,16 2,-3,5,- 1

Thus, since 97 = 24( 4) + 1, the sum of the :first

97 terms can be grouped into 24 groups of

4 terms each, with one remaining term, which

allows the sum to be easily found:

(a 1 + a 2 + a 3 + a 4 )+(as + a 6 + a 7 + a 8 ) + ... +

(a93 + a9 4 + a9s + a96) + a97

= (2-3+ 5 - 1)+(2-3+ 5 -1)+ ... +

(2-3+ 5 -1)+ 2

= 24(2- 3 + 5 -1)+ 2 = 24(3)+ 2 = 74

The correct answer is B.

PS09901

138. The sequence a 1 , a 2 , ... , an, ... is such that

an= 2an _ 1 - x for all positive integers n ::c: 2 and for a

certain number x. If a 5 = 99 and a 3 = 27, what is the

value of x?

(Al 3

(Bl 9

(Cl 18

(Dl 36

(El 45

Algebra Sequences and serie

An expression for as that involves x can be

obtained using a 3 = 27 and applying the equation

an = 2an _ 1 - x twice, once for n = 4 and once for

n =5.

= 2(27) -X

as= 2a 4 - x

using an = 2an _ 1 - x for

n = 4

using a 3 = 27

using an = 2an _ 1 - x for

n = 5

= 2[2(27) -x] -X

= 4(27) -3x

using a 4 = 2(27) -x

combine like terms

Therefore, using as= 99, we have

99 = 4(27) -3x

3x = 4(27) - 99

X = 4(9) ·- 33

x = 3

given

adding (3x - 99) to both sides

dividing both sides by 3

arithmetic

The correct answer is A.

PS03779

139. A window is in the shape of a regular hexagon with
     each side of length 80 centimeters. If a diagonal
     through the center of the hexagon is w centimeters
     long, then w =

(A) 80

(B) 120

(C) 150

(D) 160

(E) 240