Four numerical quantities K, L, M, and N are defined based on arithmetic and geometric seq… — GMAT

Four numerical quantities $K$ , $L$ , $M$ , and $N$ are defined based on arithmetic and geometric sequences as follows. Arrange these four quantities in ascending order (from smallest to largest value):

- **Quantity $K$ **: The sum of the first 5 terms of an arithmetic sequence with first term $a_1 = 2$ and common difference $d = 3$ .
- **Quantity $L$ **: The 4th term of a geometric sequence with first term $b_1 = 3$ and common ratio $r = 2$ .
- **Quantity $M$ **: The sum of an infinite geometric series with first term $c_1 = 18$ and common ratio $r = \frac{1}{2}$ .
- **Quantity $N$ **: The 7th term of an arithmetic sequence with first term $d_1 = 50$ and common difference $d = -4$ .

Which of the following represents the correct ascending order of the four quantities?

1Quantity $L$
2Quantity $N$
3Quantity $M$
4Quantity $K$

Answer

The correct ascending order of the quantities is Quantity L (24) < Quantity N (26) < Quantity M (36) < Quantity K (40).

Evaluating each quantity gives Quantity L = 24, Quantity N = 26, Quantity M = 36, and Quantity K = 40. Arranging these in ascending numerical order produces the sequence Quantity L, Quantity N, Quantity M, Quantity K.

Step-by-Step Solution

Calculate Quantity K

K = 40

The sum of an arithmetic sequence is given by

S_n = \frac{n}{2}[2a_1 + (n-1)d]

. For

n = 5

a_1 = 2

, and

d = 3

S_5 = \frac{5}{2}[2(2) + (5-1)(3)] = \frac{5}{2}[4 + 12] = \frac{5}{2}(16) = 40

Calculate Quantity L

L = 24

The

n

-th term of a geometric sequence is

b_n = b_1 \cdot r^{n-1}

. For

n = 4

b_1 = 3

, and

r = 2

b_4 = 3 \cdot 2^{4-1} = 3 \cdot 2^3 = 3 \cdot 8 = 24

Calculate Quantity M

M = 36

The sum of an infinite geometric series with

|r| < 1

S_\infty = \frac{c_1}{1 - r}

. For

c_1 = 18

and

r = \frac{1}{2}

S_\infty = \frac{18}{1 - 1/2} = \frac{18}{1/2} = 36

Calculate Quantity N

N = 26

The

n

-th term of an arithmetic sequence is

d_n = d_1 + (n-1)d

. For

n = 7

d_1 = 50

, and

d = -4

d_7 = 50 + (7-1)(-4) = 50 + 6(-4) = 50 - 24 = 26

Compare the evaluated quantities to arrange them in ascending order

24 < 26 < 36 < 40

, which corresponds to

L < N < M < K

Comparing the numeric values directly yields

24 \text{ (L)} < 26 \text{ (N)} < 36 \text{ (M)} < 40 \text{ (K)}

Key Concept

Arithmetic and Geometric Sequence Formulas

Question

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