Notice that between each of these numbers, there is a difference of \(\displaystyle 6\). This means that for each element, you will add \(\displaystyle 6\). The first element is \(\displaystyle 6-2\) or \(\displaystyle 4\). The second is \(\displaystyle 6*2-2\) or \(\displaystyle 10\), and so forth... Therefore, for the \(\displaystyle 20\)th element, the value will be \(\displaystyle 6*20-2\) or \(\displaystyle 118\).

Use the formula a_n = a₁ + (n – 1)d

a₆ = a₁ + 5d

a₉ = a₁ + 8d

Subtracting these equations yields

a₆ – a₉ = –3d

–7 – 8 = –3d

d = 5

a₁ = 33

Then use the formula for the series; = –30

The sequence is defined by a_n = 4n – 3  for such n = 1,2,3,4....

What is the next term in the following sequence?

\(\displaystyle \left \{ 4,11,18,25,32... \right \}\)

This is an arithmetic sequence with a common difference of \(\displaystyle 7\). To find the next term in an arithmetic sequence, add the common difference to the previously listed term:

\(\displaystyle 32+7=39\)

This question can be answered by analyzing the sequence provided and determining the pattern. The first term is \(\displaystyle 3\), and the second term is \(\displaystyle 7.\) The third term is \(\displaystyle 11.\)Thus, \(\displaystyle 4\) has been added to \(\displaystyle 3\) in order to obtain \(\displaystyle 7\), and \(\displaystyle 4\) has been added to \(\displaystyle 7\) in order to obtain \(\displaystyle 11.\) This shows that \(\displaystyle 4\) is added to each preceding term in the sequence in order to obtain the next term. The complete sequence from terms one through six is shown below.

\(\displaystyle 3,7,11,15,19,23\)

Thus, the sixth term is \(\displaystyle 23.\)  

Begin by looking at the transitions from number to number in this series:

From \(\displaystyle 4\) to \(\displaystyle 11\): Add \(\displaystyle 7\)

From \(\displaystyle 11\) to \(\displaystyle 9\): Subtract \(\displaystyle 2\)

From \(\displaystyle 9\) to \(\displaystyle 16\): Add \(\displaystyle 7\)

From \(\displaystyle 16\) to \(\displaystyle 14\): Subtract \(\displaystyle 2\)

From what you can tell, you can guess that the next step will be to add \(\displaystyle 7\). Thus, the next value will be \(\displaystyle 21\).

This sequence is a little tricky. Notice that the second element is equal to the first. After that, the third is equal to the sum of the first two \(\displaystyle (1+1)\), next the fourth is equal to the sum of the second and the third \(\displaystyle (2+1)\), and the same continues for each element after this. Thus, the next element in the series will be equal to \(\displaystyle 8+13\) or \(\displaystyle 21\).

The difference between each term is constant, thus the sequence is an arithmetic sequence.

Simply find the difference between each term, and add it to the last term to find the next term.

\(\displaystyle \\ 15-9 = 6 \newline 15+6 = 21\)

  Ryan is the 11^th oldest and 9^th youngest, so there are 10 students older than him and 8 younger than him. The 10 older plus the 8 younger plus Ryan himself gives a total of 10+8+1 = 19. (The trick is not to double-count Ryan.)

Each number is the previous number + 3.

Example: 1 + 3 = 4

13+3 = 16