Find the Next Term 7 , -14 , 28 , -56 , 112 , -224

Solution:

Step 1:

Identify the pattern in the sequence. Each term is obtained by multiplying the previous term by $-2$. This indicates a geometric sequence with a common ratio $r = -2$.

Step 2:

The general formula for the nth term of a geometric sequence is $a_{n} = a_{1} \cdot r^{n - 1}$.

Step 3:

Plug in the first term $a_{1} = 7$ and the common ratio $r = -2$ into the formula: $a_{n} = 7 \cdot (-2)^{n - 1}$.

Step 4:

To find the 7th term, set $n = 7$ in the formula: $a_{7} = 7 \cdot (-2)^{7 - 1}$.

Step 5:

Calculate $7 - 1$ to find the exponent for $-2$: $a_{7} = 7 \cdot (-2)^{6}$.

Step 6:

Compute $(-2)^{6}$, which is $64$: $a_{7} = 7 \cdot 64$.

Step 7:

Finally, multiply $7$ by $64$ to get the 7th term: $a_{7} = 448$.

Knowledge Notes:

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The nth term of a geometric sequence can be found using the formula $a_{n} = a_{1} \cdot r^{n - 1}$, where:

• $a_{n}$ is the nth term of the sequence,
• $a_{1}$ is the first term of the sequence,
• $r$ is the common ratio, and
• $n$ is the term number.

In this problem, the sequence is geometric with a common ratio of $-2$. This means that each term is the previous term multiplied by $-2$. To find any term in the sequence, we use the formula with the given first term and common ratio. The exponent in the formula represents the number of times the common ratio is used to reach the nth term from the first term. When dealing with negative common ratios, the sign of the terms will alternate between positive and negative. When raising a negative number to an even power, the result is positive, and when raising it to an odd power, the result is negative.