Sequences and series: แบบฝึกหัด

Find the sum of the first 8 terms of the geometric series \( 5 + 10 + 20 + \dots \)

In the binomial expansion of \( (2 + kx)^6 \), where \( k \) is a constant, the coefficient of \( x^2 \) is 60. Find the possible values of \( k \).

The first term of an arithmetic series is 5 and the common difference is 3. Given that the sum of the first \( n \) terms is 670, find the value of \( n \).

A geometric series has a sum to infinity of 8. The second term of the series is 2. Find the common ratio \( r \) of the series.

For an arithmetic series, the sum of the first \( n \) terms is given by \( S_n = 3n^2 + 2n \). Find the first term \( a \) and the common difference \( d \).

The second term of a geometric series is \( 4 \) and the sum to infinity is \( 18 \). Find the two possible values of the common ratio \( r \).

Find the smallest value of \( n \) such that the sum of the first \( n \) terms of the geometric series \( 1 + 1.2 + 1.44 + \dots \) exceeds \( 100 \).