Is my answer correct? QUESTION The sequence 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12...... can be generated by a closed formula. Find the formula and the next four terms of the sequence.

ii. Find the next three terms of the series, provide a closed formula and calculate the sum to the first n terms for the following series: 6+36+216+...................

iii. Find the next three terms of the series, provide a closed formula, and calculate the sum to the first n terms for the following series: 21+24+27+...................

MY ANSWER:

The sequence 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12... can be generated by the following closed formula:

a(n) = n + ceil(n/6)

The next four terms of the sequence are:

a(13) = 13 + ceil(13/6) = 14

a(14) = 14 + ceil(14/6) = 15

a(15) = 15 + ceil(15/6) = 16

a(16) = 16 + ceil(16/6) = 17

So, the next four terms of the sequence are 14, 15, 16, 17.

ii. The given series 6+36+216+... is a geometric series with a common ratio of 6. The closed formula for the nth term (a(n)) of a geometric series is:

a(n) = a(1) * r^(n-1)

In this series, a(1) = 6 and r = 6. Therefore, the closed formula for the nth term is:

a(n) = 6 * 6^(n-1)

To find the next three terms of the series, we can substitute n = 5, 6, and 7 into the formula:

a(5) = 6 * 6^(5-1) = 6 * 6^4 = 6 * 1296 = 7776

a(6) = 6 * 6^(6-1) = 6 * 6^5 = 6 * 7776 = 46656

a(7) = 6 * 6^(7-1) = 6 * 6^6 = 6 * 46656 = 279936

The next three terms of the series are 7776, 46656, 279936.

To calculate the sum to the first n terms of the series, we can use the formula for the sum of a geometric series:

S(n) = a(1) * (1 - r^n) / (1 - r)

In this series, a(1) = 6 and r = 6. So, the closed formula for the sum to the first n terms is:

S(n) = 6 * (1 - 6^n) / (1 - 6)

iii. The given series 21+24+27+... is an arithmetic series with a common difference of 3. The closed formula for the nth term (a(n)) of an arithmetic series is:

a(n) = a(1) + (n - 1) * d