Determine the 10th term of the sequence 3,10,28,72,176. state the general term

n = 1

a1 = 3 = 1 * 3 = 1 * ( 2 * 1 + 1 ) = 2 ^ 0 ( 2 n + 1 ) = 2 ^ ( n - 1 ) ( 2 n + 1 )

n = 2

a2 = 10 = 2 * 5 = 2 * ( 2 * 2 + 1 ) = 2 ^ 1 ( 2 n + 1 ) = 2 ^ ( n - 1 ) ( 2 n + 1 )

n = 3

a3 = 28 = 4 * 7 = 4 * ( 2 * 3 + 1 ) = 2 ^ 2 ( 2 n + 1 ) = 2 ^ ( n - 1 ) ( 2 n + 1 )

n = 4

a4 = 72 = 8 * 9 = 8 * ( 2 * 4 + 1 ) = 2 ^ 3 ( 2 n + 1 ) = 2 ^ ( n - 1 ) ( 2 n + 1 )

n = 5

a5 = 176 = 16 * 11 = 16 * ( 2 * 5 + 1 ) = 2 ^ 4 ( 2 n + 1 ) = 2 ^ ( n - 1 ) ( 2 n + 1 )

SO :

an = 2 ^ ( n - 1 ) ( 2 n + 1 )

a10 = 2 ^ ( 10 - 1 ) * ( 2 * 10 + 1 ) =

2 ^ 9 * ( 20 + 1 ) = 512 * 21 = 10752

To find the 10th term of a sequence, we need to determine the pattern and formula that generate the terms.

Looking at the given sequence: 3, 10, 28, 72, 176, we can observe that each term is increasing as follows:

1st term: 3
2nd term: 10 = 3 + (3 * 2 + 1)
3rd term: 28 = 10 + (10 * 2 + 8)
4th term: 72 = 28 + (28 * 2 + 16)
5th term: 176 = 72 + (72 * 2 + 32)

From this pattern, we can deduce that each term is obtained by adding the result of a term multiplied by 2, and then adding a specific number.

So, we can express the general term of this sequence as follows:

Term(n) = Term(n-1) + (Term(n-1) * 2 + X)

Where X is a constant difference between each term.

To determine X, we can calculate the differences between consecutive terms:

Difference between 2nd and 1st term: 10 - 3 = 7
Difference between 3rd and 2nd term: 28 - 10 = 18
Difference between 4th and 3rd term: 72 - 28 = 44
Difference between 5th and 4th term: 176 - 72 = 104

We can observe that the differences are increasing by a factor of 2 each time. This implies that X should also increase by a factor of 2 each time.

Now, let's calculate X:

Difference of the 2nd term: 7
Difference of the 3rd term: 7 * 2 = 14
Difference of the 4th term: 14 * 2 = 28
Difference of the 5th term: 28 * 2 = 56

Since the differences are increasing by a factor of 2, X should be equal to half of the difference between the 4th and 3rd term:

X = (72 - 28) / 2
X = 44 / 2
X = 22

Now that we have X, we can find the 10th term by recursively applying our general term formula:

Term(10) = Term(10-1) + (Term(10-1) * 2 + 22)
= Term(9) + (Term(9) * 2 + 22)
= ...

To simplify the process, we can start from the 1st term and work our way up to the 10th term, using the formula Term(n) = Term(n-1) + (Term(n-1) * 2 + 22):

1st term: 3
2nd term: 3 + (3 * 2 + 22) = 3 + 6 + 22 = 31
3rd term: 31 + (31 * 2 + 22) = 31 + 62 + 22 = 115
4th term: 115 + (115 * 2 + 22) = 115 + 230 + 22 = 367
5th term: 367 + (367 * 2 + 22) = 367 + 734 + 22 = 1123
6th term: 1123 + (1123 * 2 + 22) = 1123 + 2246 + 22 = 3391
7th term: 3391 + (3391 * 2 + 22) = 3391 + 6782 + 22 = 10195
8th term: 10195 + (10195 * 2 + 22) = 10195 + 20390 + 22 = 30607
9th term: 30607 + (30607 * 2 + 22) = 30607 + 61214 + 22 = 91843
10th term: 91843 + (91843 * 2 + 22) = 91843 + 183686 + 22 = 275551

Therefore, the 10th term of the sequence is 275551.

The general term of the sequence is Term(n) = Term(n-1) + (Term(n-1) * 2 + 22).