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Difficulty: HardArithmetic and Geometric Sequences and Series

Sequence AA is an arithmetic sequence with first term a1=5a_1 = 5 and common difference d=3d = 3.
Sequence BB is a geometric sequence with first term b1=2b_1 = 2 and common ratio r=3r = \sqrt{3}.

Arrange the four quantities defined below in ascending order (from smallest to largest value).

  1. 1Quantity K: The 15th term of Sequence A
  2. 2Quantity N: The sum of the first 6 terms of Sequence B
  3. 3Quantity M: The 9th term of Sequence B
  4. 4Quantity L: The sum of the first 10 terms of Sequence A

Answer

The correct ascending order is Quantity K, followed by Quantity N, Quantity M, and finally Quantity L.
Evaluating each expression gives Quantity K = 47, Quantity N ≈ 142.07, Quantity M = 162, and Quantity L = 185. Comparing these values from smallest to largest yields the order K, N, M, L.

Step-by-Step Solution

1
Calculate Quantity K (a15a_{15} for Sequence A)
a15=5+(15−1)×3=47a_{15} = 5 + (15 - 1) \times 3 = 47
The nn-th term of an arithmetic sequence is given by an=a1+(n−1)da_n = a_1 + (n - 1)d.
2
Calculate Quantity L (S10S_{10} for Sequence A)
S10=102×[2(5)+(10−1)×3]=5×(10+27)=185S_{10} = \frac{10}{2} \times [2(5) + (10 - 1) \times 3] = 5 \times (10 + 27) = 185
The sum of the first nn terms of an arithmetic sequence is given by Sn=n2[2a1+(n−1)d]S_n = \frac{n}{2}[2a_1 + (n - 1)d].
3
Calculate Quantity M (b9b_9 for Sequence B)
b9=2×(3)9−1=2×(3)8=2×34=162b_9 = 2 \times (\sqrt{3})^{9 - 1} = 2 \times (\sqrt{3})^8 = 2 \times 3^4 = 162
The nn-th term of a geometric sequence is given by bn=b1rn−1b_n = b_1 r^{n-1}.
4
Calculate and approximate Quantity N (S6S_6 for Sequence B)
S6=2((3)6−1)3−1=2(27−1)3−1=523−1=52(3+1)≈52×2.732=142.07S_6 = \frac{2((\sqrt{3})^6 - 1)}{\sqrt{3} - 1} = \frac{2(27 - 1)}{\sqrt{3} - 1} = \frac{52}{\sqrt{3} - 1} = 52(\sqrt{3} + 1) \approx 52 \times 2.732 = 142.07
The sum of a geometric series is Sn=b1(rn−1)r−1S_n = \frac{b_1(r^n - 1)}{r - 1}. Rationalizing the denominator yields 52(3+1)52(\sqrt{3} + 1).
5
Compare all four values to establish ascending order
47<142.07<162<18547 < 142.07 < 162 < 185, which corresponds to K<N<M<LK < N < M < L
Ordering the numerical outputs from least to greatest gives the required sequence.

Key Concept

Calculating specific terms and sums of arithmetic and geometric sequences using explicit formulas and ordering calculated quantities.
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Topics

Arithmetic and Geometric Sequences and Series

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