Sequences & Series: Level 1
1.
The first term in a sequence is 3, and each subsequent term is 4 more than the term before it. What is the value of the 11th term?
Solution (C): This is an arithmetic sequence with \(a_1 = 3\) and \(d = 4\).
The formula is \(a_n = a_1 + (n-1)d\).
\(a_{11} = 3 + (11-1) \times 4 = 3 + (10 \times 4) = 3 + 40 = 43\).
2.
In the sequence 5, 10, 20, 40, … each term after the first is 2 times the previous term. What is the 6th term?
Solution (B): This is a geometric sequence.
5th term: \(40 \times 2 = 80\).
6th term: \(80 \times 2 = 160\).
Alternatively, \(a_n = a_1 \times r^{(n-1)} = 5 \times 2^{(6-1)} = 5 \times 2^5 = 5 \times 32 = 160\).
3.
What is the sum of all integers from 1 to 100, inclusive?
Solution (D): This is an arithmetic series with 100 terms.
Use the sum formula: \(S_n = \frac{n}{2}(a_1 + a_n)\).
\(S_{100} = \frac{100}{2}(1 + 100) = 50(101) = 5050\).
4.
The sum of three consecutive integers is 30. What is the value of the largest of these integers?
Solution (C): For a set of consecutive integers, the average is the middle number.
Average = \(30 \div 3 = 10\).
So, 10 is the middle integer. The integers are 9, 10, 11.
The largest is 11.
5.
The first term of a sequence is 81. Each term after the first is one-third of the previous term. What is the 4th term?
Solution (B): This is a geometric sequence.
\(a_1 = 81\)
\(a_2 = 81 \div 3 = 27\)
\(a_3 = 27 \div 3 = 9\)
\(a_4 = 9 \div 3 = 3\)
6.
What is the sum of the multiples of 5 from 5 to 50, inclusive?
Solution (C): This is an arithmetic series with \(a_1 = 5\) and \(a_n = 50\).
1. Find the number of terms \(n\): \(\frac{a_n – a_1}{d} + 1 = \frac{50 – 5}{5} + 1 = \frac{45}{5} + 1 = 9 + 1 = 10\) terms.
2. Find the sum: \(S_n = \frac{n}{2}(a_1 + a_n) = \frac{10}{2}(5 + 50) = 5(55) = 275\).
7.
A sequence is defined by \(S_1 = 3\) and \(S_n = S_{n-1} + 2\) for \(n > 1\). What is the value of \(S_{21}\)?
Solution (D): This is an arithmetic sequence with first term \(a_1 = 3\) and common difference \(d = 2\). We need to find the 21st term.
Formula: \(a_n = a_1 + (n-1)d\).
\(a_{21} = 3 + (21-1) \times 2 = 3 + (20 \times 2) = 3 + 40 = 43\).
8.
What is the sum of the series \(1 + 3 + 3^2 + 3^3\)?
Solution (C): This is a simple sum.
\(1 + 3 + 9 + 27\)
\(4 + 9 + 27 = 13 + 27 = 40\).
9.
What is the sum of the positive even integers from 2 to 20, inclusive?
Solution (B): This is an arithmetic series with \(a_1 = 2\), \(a_n = 20\), and \(d = 2\).
1. Find the number of terms \(n\): \(\frac{20 – 2}{2} + 1 = \frac{18}{2} + 1 = 9 + 1 = 10\) terms.
2. Find the sum: \(S_n = \frac{n}{2}(a_1 + a_n) = \frac{10}{2}(2 + 20) = 5(22) = 110\).
10.
The sum of 5 consecutive even integers is 50. What is the value of the smallest integer?
Solution (B): The average of the 5 integers is the middle (3rd) integer.
Average = \(50 \div 5 = 10\).
The 3rd integer is 10.
The integers are {6, 8, 10, 12, 14}.
The smallest integer is 6.
Score: 0 / 10