Question

Find the fourth term of a GP with first term 7 and common ratio -4

Short answer

The fourth term of the geometric progression is -448.

Step-by-step solution

Identify first term and common ratio

In a geometric progression (GP), the first term (denoted as a) is given as 7 and the common ratio (denoted as r) is given as -4.

Use the nth term formula of a GP

The nth term (T_n) of a geometric progression can be found using the formula: T_n = a * r^(n - 1). Here, the desired term is the fourth term (n=4).

Calculate the fourth term

Substitute the known values into the formula to find the fourth term (T_4): T_4 = 7 * (-4)^(4 - 1) = 7 * (-4)^3.

Simplify the expression

Calculate the exponent and multiply by the first term: T_4 = 7 * (-64) = -448.

Key concepts

nth term formula

Understanding the nth term formula is crucial when studying geometric progressions (GP). The formula enables us to find any term in the sequence without having to list all the previous terms.

The nth term of a GP is given by the formula:
\( T_n = a \cdot r^{(n - 1)} \),
where \( T_n \) is the nth term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. In the given exercise, we use this formula to directly calculate the fourth term. Simply plug in the first term (7), the common ratio (-4), and the term number (4) to find the fourth term: \( T_4 = 7 \cdot (-4)^3 \). This formula is especially powerful for large values of \( n \), where manually computing each term would be impractical.

common ratio

The common ratio, often denoted as \( r \), is the factor by which consecutive terms of a GP are multiplied to get the next term. This constant factor plays a pivotal role in determining the nature of the progression, whether it's increasing, decreasing, alternating, or approaching zero.

It's calculated by dividing any term in the sequence by the preceding term (except the first, which has no predecessor). For our example, \( r = -4 \), indicating an alternating sequence where each term changes sign from positive to negative or vice versa. Understanding the common ratio is essential, as it influences not only individual term values but also the overall behavior of the sequence.

exponents in GP

When dealing with a geometric progression, exponents become a key focal point because each term is derived by raising the common ratio to an exponent that reflects its position in the sequence. The exponent indicates how many times the common ratio multiplies the first term.

In the exercise, the exponent in the fourth term is \( 3 \) since the formula requires subtracting \( 1 \) from the term number: \( (-4)^{4 - 1} = (-4)^3 \). These exponents highlight the power of geometric progressions to rapidly increase or decrease and also to alternate based on whether the common ratio is positive, negative, greater than one, less than one, or a fraction.