Series：练习题

The first term of an arithmetic progression is \( 8 \) and the common difference is \( -3 \). Find the \( 15 \)th term of the progression.

The first term of an arithmetic progression is 10 and the 15th term is 52. Calculate the sum of the first 15 terms.

In the expansion of \( (1 + ax)^n \) in ascending powers of \( x \), the first three terms are \( 1 + 24x + 252x^2 \). Find the values of the constants \( a \) and \( n \).

Find the sum of the first 20 terms of an arithmetic progression where the first term is 5 and the common difference is 3.

The second term of a geometric progression is 6 and the sum to infinity is 32. Find the possible values of the first term \( a \).

Find the coefficient of \( x^2 \) in the expansion of \( \left( 2x - \frac{1}{x} \right)^6 \).

The common ratio of a geometric progression is \( r \). The first term is 5. Given that the sum of the first three terms is 4.05, find the possible values of \( r \).

A geometric progression has first term \( a \) and common ratio \( r \). The sum of the first two terms is 15 and the sum to infinity is 27. Find the two possible values of \( r \).

An arithmetic progression has first term \( a \) and common difference \( d \). The sum of the first 10 terms is 310.

(a) Show that \( 2a + 9d = 62 \).

The 2nd, 7th, and 25th terms of this arithmetic progression are the first three terms of a geometric progression.

(b) Find the value of \( d \) in terms of \( a \).
(c) Given that \( d \neq 0 \), find the values of \( a \) and \( d \).

(a) Find the first three terms in the expansion of \( (1 + px)^n \) in ascending powers of \( x \), where \( n \) is a positive integer and \( p \) is a constant.

(b) In the expansion of \( (1 + px)^n \), the coefficient of \( x \) is \( -24 \) and the coefficient of \( x^2 \) is \( 252 \). Form two equations in \( n \) and \( p \) and solve them to find the values of \( n \) and \( p \).

(c) Using the values of \( n \) and \( p \) found in part (b), find the coefficient of \( x^3 \) in the expansion of \( (1 - 2x)(1 + px)^n \).