Question

How many terms are there in the sequence 1, 8, 28, 56, ..., 1 ?

Answer 1

Without a clear pattern, we count the terms from the first '1' to the last '1' in the provided sequence, resulting in a total of 17 terms.

Step-by-step explanation:

The sequence provided appears to be a list of separate numbers with no immediately discernible pattern. To determine the number of terms in the sequence ending with '1', simply count each term given. However, as the sequence is listed as '1, 8, 28, 56, ..., 1', it could be interpreted that we should count from the first '1' to the last '1'. Without a clear pattern or rule for the sequence, we cannot use a formula and must rely on the explicit listing of terms. Therefore, the count includes all terms from the beginning to the end.

In the list provided, there are 17 terms, so if we interpret the sequence to start and end with '1', we would say there are 17 terms.

Answer 2

9 terms

Step-by-step explanation:

Given:

1, 8, 28, 56, ..., 1

Required

Determine the number of sequence

To determine the number of sequence, we need to understand how the sequence are generated

The sequence are generated using

`\left[\begin{array}{c}n&&r\end{array}\right] = (n!)/((n-r)!r!)`

Where n = 8 and r = 0,1....8

When r = 0

`\left[\begin{array}{c}8&&0\end{array}\right] = (8!)/((8-0)!0!) = (8!)/(8!0!) = 1`

When r = 1

`\left[\begin{array}{c}8&&1\end{array}\right] = (8!)/((8-1)!1!) = (8!)/(7!1!) = (8 * 7!)/(7! * 1) = (8)/(1) = 8`

When r = 2

`\left[\begin{array}{c}8&&2\end{array}\right] = (8!)/((8-2)!2!) = (8!)/(6!2!) = (8 * 7 * 6!)/(6! * 2 *1) = (8 * 7)/(2 *1) =2 8`

When r = 3

`\left[\begin{array}{c}8&&3\end{array}\right] = (8!)/((8-3)!3!) = (8!)/(5!3!) = (8 * 7 * 6 * 5!)/(5! *3* 2 *1) = (8 * 7 * 6)/(3 *2 *1) = 56`

When r = 4

`\left[\begin{array}{c}8&&4\end{array}\right] = (8!)/((8-4)!4!) = (8!)/(4!3!) = (8 * 7 * 6 * 5 * 4!)/(4! *4*3* 2 *1) = (8 * 7 * 6*5)/(4*3 *2 *1) = 70`

When r = 5

`\left[\begin{array}{c}8&&5\end{array}\right] = (8!)/((8-5)!5!) = (8!)/(5!3!) = (8 * 7 * 6 * 5!)/(5! *3* 2 *1) = (8 * 7 * 6)/(3 *2 *1) = 56`

When r = 6

`\left[\begin{array}{c}8&&6\end{array}\right] = (8!)/((8-6)!6!) = (8!)/(6!2!) = (8 * 7 * 6!)/(6! * 2 *1) = (8 * 7)/(2 *1) = 28`

When r = 7

`\left[\begin{array}{c}8&&7\end{array}\right] = (8!)/((8-7)!7!) = (8!)/(7!1!) = (8 * 7!)/(7! * 1) = (8)/(1) = 8`

When r = 8

`\left[\begin{array}{c}8&&8\end{array}\right] = (8!)/((8-8)!8!) = (8!)/(8!0!) = 1`

The full sequence is: 1,8,28,56,70,56,28,8,1

And the number of terms is 9