Arithmetic and geometric sequences and their summations：練習問題

In a geometric sequence, the 3rd term is $$18$$ and the 6th term is $$486$$. Find the common ratio of the sequence.

The sum of the first $$n$$ terms of an arithmetic sequence is given by $$S_n = 2n^2 + 3n$$. Find the 5th term of the sequence.

A geometric sequence is denoted by \(x_1, x_2, x_3, \dots\) with common ratio \(r\). The sum to infinity of this sequence is 24. It is given that the first term, twice the second term, and three times the third term, i.e., the terms \(x_1, 2x_2, 3x_3\), form an arithmetic sequence. Find the value of \(r\).

The first term of an arithmetic sequence is \(5\) and the common difference is \(3\). What is the \(10\)-th term of the sequence?

Find the sum of the first 5 terms of the geometric sequence $$2, 6, 18, \dots$$

In an arithmetic sequence, the first term is \(15\) and the common difference is \(-2\). Find the \(8^{\text{th}}\) term of the sequence.

An object travels \(10\) meters in the first second. In each subsequent second, it travels \(5\% \) less distance than the previous second. Find the total distance (in meters) the object travels before theoretically coming to rest (i.e., the sum to infinity).

The sum to infinity of a geometric sequence is 18, and its second term is 8. Find the common ratio and the first term of the sequence.

Consider the sequence $$5, 8, 11, \dots$$

(a) Determine whether the sequence is an arithmetic sequence or a geometric sequence. State its common difference or common ratio.

(b) Find the $$10^{th}$$ term of the sequence.

(c) Find the sum of the first $$6$$ terms of the sequence.

An arithmetic sequence, denoted by \(A\), has the first term \(a_1\) and common difference \(d\). A geometric sequence, denoted by \(G\), has the first term \(g_1\) and common ratio \(r\).

(a) It is given that \(a_1 = g_1\). Let \(a = a_1 = g_1\). The sum of the first 10 terms of sequence \(A\), \(S_{10, A}\), is \(-10\). The 4th term of sequence \(A\) is equal to the 3rd term of sequence \(G\). The sum to infinity of sequence \(G\), \(S_{\infty, G}\), is \(16\).

• (i) Express \(a\) in terms of \(r\).
• (ii) Show that \(1 - r^2 = -\frac{3}{a} \left( \frac{2a + 10}{9} \right)\).
• (iii) Hence, find the common ratio \(r\) of sequence \(G\).

(b) Using the value of \(r\) found in (a)(iii), find the common difference \(d\) of sequence \(A\).