1.
- Find the common difference and the next term
of the following sequence:
Correct Answer
A. 8
Explanation
The common difference in this sequence is 8 because each term is obtained by adding 8 to the previous term. The next term can be found by adding 8 to the last term, which is 35. Therefore, the next term in the sequence is 43.
2.
- Find the common ratio and the seventh term of
the following sequence:
Correct Answer
B. 3
Explanation
The common ratio of the given sequence is 3 because each term is obtained by multiplying the previous term by 3. To find the seventh term, we start with the first term (2/9) and multiply it by the common ratio 6 times (since we want the seventh term). So, the seventh term of the sequence is 2/9 * 3 * 3 * 3 * 3 * 3 * 3 = 486/9 = 54.
3.
The first term of an arithmetic sequence is 10, and the common difference is 4. Find the 12th term of the sequence.
Correct Answer
C. 54
Explanation
In an arithmetic sequence, each term is obtained by adding a constant value to the previous term. In this specific sequence, the first term is 10, and each subsequent term increases by 4. To find the 12th term, we use the formula which considers the position of the term (n), and after calculation, we find that the 12th term is 54.
4.
- Find the tenth term and the n-th
term of the following sequence:
Correct Answer
D. 256
Explanation
The given sequence is a geometric sequence with a common ratio of 2. To find the tenth term, we can use the formula for the nth term of a geometric sequence: aₙ = a₁ * r^(n-1). Plugging in the values, we get a₁ = 1/2 and r = 2. So, the tenth term is a₁₀ = (1/2) * 2^(10-1) = 256. Therefore, the correct answer is 256.
5.
Find the first term of the arithmetic sequence having a_{4} = 93 and a_{8} = 65.
Correct Answer
C. 114
Explanation
The arithmetic sequence is formed by adding a common difference to each term. To find the common difference, we can subtract a4 from a8: 65 - 93 = -28. Now we can find the first term by subtracting three times the common difference from a4: 93 - 3(-28) = 177. Therefore, the first three terms of the sequence are 177, 149, and 121. The n-th term can be found by adding (n-1) times the common difference to the first term. So, the n-th term is 177 + (n-1)(-28) = 205 - 28n. Plugging in n = 4, we get 205 - 28(4) = 205 - 112 = 93, which matches a4. Therefore, the correct answer is 114.
6.
- Find the n-th
and the 26th
terms of the geometric sequence with a_{5}
= 5/4 and a_{12}
= 160.
Correct Answer
A. 2 621 440
Explanation
The given geometric sequence has a common ratio of 4. To find the nth term, we can use the formula an = a1 * r^(n-1), where a1 is the first term and r is the common ratio. We are given a5 = 5/4, so we can substitute this value to find a1 * r^4 = 5/4. Similarly, a12 = 160 can be substituted to find a1 * r^11 = 160. Solving these two equations, we find that a1 = 5/32 and r = 2. Substituting n = 26 into the formula, we get a26 = (5/32) * 2^(26-1) = 2,621,440. Therefore, the 26th term is 2,621,440.
7.
- Find the sum of 1 + 5 + 9 + ... + 49 + 53.
Correct Answer
378
8.
Find a_{n} if S_{4} = 26/27 and r = 1/3.
Correct Answer
13/20
9.
Differentiate arithmetic and geometric sequence.