Numerical Sequence Patterns - SSAT Upper Level Quantitative

1. Consider the sequence $2, 6, 18, 54, 162, \ldots$ and the sequence $1, 5, 25, 125, 625, \ldots$. If the $n$th terms of these sequences are added together, what type of sequence is formed?
   Answer: Neither arithmetic nor geometric The first sequence is $2 \cdot 3^{n-1}$ (geometric with ratio $3$). The second sequence is $5^{n-1}$ (geometric with ratio $5$). The sum sequence is $2 \cdot 3^{n-1} + 5^{n-1}$: $3, 11, 43, 179, 787, \ldots$. Differences: $8, 32, 136, 608, \ldots$ (not constant). Ratios: $\frac{11}{3} \approx 3.67$, $\frac{43}{11} \approx 3.91$, $\frac{179}{43} \approx 4.16$ (not constant). The sequence is neither arithmetic nor geometric. Choices A and D assume it's arithmetic. Choice B assumes it's geometric.
2. A sequence is defined by $a_1 = 2$, $a_2 = 5$, and $a_n = 3a_{n-1} - 2a_{n-2}$ for $n \geq 3$. Which term first exceeds $100$?
   Answer: The 7th term Computing the sequence: $a_1 = 2$, $a_2 = 5$, $a_3 = 3(5) - 2(2) = 11$, $a_4 = 3(11) - 2(5) = 23$, $a_5 = 3(23) - 2(11) = 47$, $a_6 = 3(47) - 2(23) = 95$, $a_7 = 3(95) - 2(47) = 191$. The 7th term is the first to exceed $100$. Choice A gives $a_6 = 95 < 100$. Choices C and D are later terms that also exceed $100$ but are not the first.
3. In a sequence, the first term is $3$, and each subsequent term is obtained by multiplying the previous term by $2$ and then subtracting $1$. What is the difference between the 5th term and the 4th term?
   Answer: $16$ Build the sequence by multiplying the previous term by 2 and then subtracting 1: $a_1 = 3$, $a_2 = 2(3) - 1 = 5$, $a_3 = 2(5) - 1 = 9$, $a_4 = 2(9) - 1 = 17$, $a_5 = 2(17) - 1 = 33$. The difference is $a_5 - a_4 = 33 - 17 = 16$. A quick check: the gaps between terms are 2, 4, 8, 16, doubling each time, because $a_{n+1} - a_n = 2(a_n - a_{n-1})$. Answers of 22 and 23 come from an arithmetic slip in building or subtracting the terms; recheck that $2(17) - 1 = 33$. Answers of 45 and 47 are larger than the 5th term itself, so they cannot be the difference between two positive terms.
4. A geometric sequence has its 3rd term equal to $18$ and its 6th term equal to $486$. If all terms are positive, what is the 4th term?
   Answer: $54$ When you encounter geometric sequence problems, remember that each term is found by multiplying the previous term by a constant ratio $r$. The general formula is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term. Given that the 3rd term is $18$ and the 6th term is $486$, you can write: - $a_3 = a_1 \cdot r^2 = 18$ - $a_6 = a_1 \cdot r^5 = 486$ To find the common ratio, divide the second equation by the first: $\frac{a_6}{a_3} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = r^3 = \frac{486}{18} = 27$ Taking the cube root: $r = 3$ Now you can find $a_1$: $18 = a_1 \cdot 3^2 = 9a_1$, so $a_1 = 2$ Therefore, $a_4 = a_1 \cdot r^3 = 2 \cdot 3^3 = 2 \cdot 27 = 54$. The answer is A. Looking at the wrong choices: B) $36$ would result from incorrectly calculating $r = 2$ instead of $3$. C) $72$ might come from miscalculating $a_1 = 8$ instead of $2$. D) $81$ would result from thinking $a_4 = r^4 = 3^4$, forgetting to include the first term coefficient. Strategy tip: In geometric sequence problems, always use the ratio between given terms to find $r$ first. When terms aren't consecutive, remember that $\frac{a_m}{a_n} = r^{m-n}$. This approach avoids having to solve for $a_1$ until the end.
5. A sequence begins: $2, 7, 17, 32, 52, \ldots$ If this pattern continues, which expression represents the $n$th term for $n \geq 3$?
   Answer: $2.5n^2 - 2.5n + 2$ When you encounter a sequence problem, you need to identify the pattern by examining the differences between consecutive terms. This sequence tests your ability to recognize when a quadratic expression generates the terms. Let's find the pattern by calculating the first and second differences: First differences: $7-2=5$, $17-7=10$, $32-17=15$, $52-32=20$ Second differences: $10-5=5$, $15-10=5$, $20-15=5$ Since the second differences are constant (5), this confirms the sequence follows a quadratic pattern of the form $an^2 + bn + c$. To verify the correct answer, test choice D with the given terms: - For $n=1$: $2.5(1)^2 - 2.5(1) + 2 = 2.5 - 2.5 + 2 = 2$ ✓ - For $n=2$: $2.5(4) - 2.5(2) + 2 = 10 - 5 + 2 = 7$ ✓ - For $n=3$: $2.5(9) - 2.5(3) + 2 = 22.5 - 7.5 + 2 = 17$ ✓ Choice A gives incorrect values: for $n=1$, $5(1) - 13(1) + 10 = 2$, but for $n=2$, $5(4) - 13(2) + 10 = 16 \neq 7$. Choice B similarly fails: for $n=2$, $4(4) - 11(2) + 9 = 11 \neq 7$. Choice C also produces wrong values: for $n=2$, $3(4) - 8(2) + 7 = 7$, but for $n=3$, $3(9) - 8(3) + 7 = 10 \neq 17$. Strategy tip: When you see constant second differences, you know you're dealing with a quadratic sequence. Always test your answer choice with multiple terms from the sequence to confirm it works consistently.

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