What is the 7th term of the geometric sequence 4, 20, 100?

To find the 7th term of the geometric sequence given by 4, 20, 100, we first need to determine the common ratio of the sequence.

A geometric sequence is defined by a starting term and a common ratio, which is the factor by which we multiply each term to get to the next term. In this sequence:

• First term (a) = 4
• Second term (ar) = 20
• Third term (ar²) = 100

To find the common ratio (r), we can divide the second term by the first term:

r = 20 / 4 = 5

Now that we have the common ratio, we can also verify it by checking the third term:

r = 100 / 20 = 5

Since both calculations give us the same common ratio, we can confidently say that r = 5.

The formula for the nth term of a geometric sequence is given by:

T_n = a * r^n-1

Now, to find the 7th term (T₇):

T₇ = 4 * 5^7-1

T₇ = 4 * 5⁶

Calculating 5⁶: 5 * 5 = 25
25 * 5 = 125
125 * 5 = 625
625 * 5 = 3125

So, 5⁶ = 15625.

Now, we can calculate T₇:

T₇ = 4 * 15625 = 62500.

Therefore, the 7th term of the geometric sequence is 62500.