A geometric sequence begins 6 , 42 , 294 , … Write down the value of the first term and the common ratio.

Given geometric sequence is 6 , 42 , 294 , … To find: First term, and Common ratio. Since the given sequence is a geometric sequence And, a geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. Let the nth term of the given geometric sequence be denoted by u n . Then, u 1 = 6 , u 2 = 42 Thus, according to the definition of the common ratio r , we have r = u 2 u 1 = 42 6 = 7 And, u 1 = First term of the sequence. Hence, first term of the sequence = 6 , and common ratio = 7 .

A geometric sequence begins 6 , 42 , 294 , … Write down a recursive formula linking the nth term to the n + 1 th term.

Given geometric sequence is 6 , 42 , 294 , … To find: The recursive formula linking nth term to the n + 1 th term. Since the given sequence is a geometric sequence And, a geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. Let the nth term of the given geometric sequence be denoted by u n . Then, u 1 = 6 , u 2 = 42 Thus, according to the definition of the common ratio r , we have r = u 2 u 1 = 42 6 = 7 And, u 1 = First term of the sequence. Hence, first term of the sequence = 6 , and common ratio = 7 . We know that a geometric sequence with first term a and common ratio r has recursive formula, u n + 1 = r u n ' with u 1 = a Thus, the recursive formula for the given geometric sequence is u n + 1 = 7 u n , with u 1 = 6

A geometric sequence begins 6 , 42 , 294 , … Write down an explicit formula for the nth term.

Given geometric sequence is 6 , 42 , 294 , … To find: The explicit formula for the nth term. Since the given sequence is a geometric sequence And, a geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. Let the nth term of the given geometric sequence be denoted by u n . Then, u 1 = 6 , u 2 = 42 Thus, according to the definition of the common ratio r , we have r = u 2 u 1 = 42 6 = 7 And, u 1 = First term of the sequence. Hence, first term of the sequence = 6 , and common ratio = 7 . We know that a geometric sequence with first term a and common ratio r has an explicit formula, u n = a r n - 1 for the nth term. Thus, the explicit formula for the nth term of the given geometric sequence is u n = 6 7 n - 1 .

A geometric sequence begins 6 , 3 , 1 . 5 , … Write down the value of the first term and the common ratio.

Given geometric sequence is 6 , 3 , 1 . 5 , … To find: The value of the first term, and the common ratio. Since the given sequence is a geometric sequence And, a geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. Let the nth term of the given geometric sequence be denoted by u n . Then, u 1 = 6 , u 2 = 3 Thus, according to the definition of the common ratio r , we have r = u 2 u 1 = 3 6 = 1 2 And, u 1 = First term of the sequence. Hence, first term of the sequence = 6 , and common ratio = 1 2 .