Four sequence-derived values S_1, S_2, S_3, and S_4 are defined below. Arrange these value… — GMAT

Four sequence-derived values $S_1, S_2, S_3,$ and $S_4$ are defined below. Arrange these values in order from smallest to largest.

1 $S_1$ : The sum of the first $5$ terms of an arithmetic sequence with first term $a_1 = 3$ and common difference $d = 4$
2 $S_3$ : The $10\text{th}$ term of an arithmetic sequence with first term $c_1 = 10$ and common difference $d = 6$
3 $S_4$ : The sum of an infinite geometric series with first term $g_1 = 48$ and common ratio $r = \frac{1}{3}$
4 $S_2$ : The sum of the first $4$ terms of a geometric sequence with first term $b_1 = 2$ and common ratio $r = 3$

The correct order from smallest to largest is

S_1 < S_3 < S_4 < S_2

(corresponding to

55 < 64 < 72 < 80

Computing each value yields

S_1 = 55

S_2 = 80

S_3 = 64

, and

S_4 = 72

. Arranging these values in ascending order results in

55 < 64 < 72 < 80

, corresponding to

S_1 < S_3 < S_4 < S_2

Calculate the value of

S_1

S_1 = 55

Using the sum formula for an arithmetic sequence

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

, we have

S_5 = \frac{5}{2}[2(3) + (5-1)4] = \frac{5}{2}[6 + 16] = 55

Calculate the value of

S_2

S_2 = 80

Using the sum formula for a finite geometric sequence

S_n = \frac{a_1(r^n - 1)}{r - 1}

, we have

S_4 = \frac{2(3^4 - 1)}{3 - 1} = \frac{2(80)}{2} = 80

Calculate the value of

S_3

S_3 = 64

Using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n-1)d

, we have

c_{10} = 10 + (10-1)6 = 10 + 54 = 64

Calculate the value of

S_4

S_4 = 72

Using the sum formula for an infinite geometric series

S_\infty = \frac{a_1}{1 - r}

, we have

S_4 = \frac{48}{1 - 1/3} = \frac{48}{2/3} = 72

Order the computed values from smallest to largest

S_1 (55) < S_3 (64) < S_4 (72) < S_2 (80)

Comparing the numerical values gives

55 < 64 < 72 < 80

Evaluation and comparison of finite and infinite arithmetic and geometric sequence terms and sums

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