Question

Find the sixth term of a GP with first term -3 and common ratio 5

Short answer

-9375

Step-by-step solution

Identify the First Term

The first term (denoted as a) of the geometric progression (GP) is given as -3. This is the starting point of the series.

Identify the Common Ratio

The common ratio (denoted as r) of the GP is given as 5. This is the factor by which we multiply each term to get the next term in the series.

Use the Formula for the nth Term

For a geometric progression, the nth term is given by the formula: \( a_n = a r^{(n-1)} \), where \( a_n \) is the nth term. We substitute \( a = -3 \), \( r = 5 \), and \( n = 6 \) into the formula.

Calculate the Sixth Term

Substitute the values into the formula to find the sixth term: \( a_6 = (-3) \times 5^{(6-1)} = (-3) \times 5^5 = (-3) \times 3125 = -9375 \).

Key concepts

Understanding the First Term of a Geometric Progression

The first term in a geometric progression, denoted as 'a', serves as the anchor point from which the sequence begins and develops. In our example, the first term is -3. It's crucial for students to grasp that this initial value is used in conjunction with the common ratio to generate all subsequent terms of the sequence. Recognizing and accurately identifying the first term is the first step in solving problems involving geometric progressions.

So, if we visualize a geometric progression as a line of dominos, the first term would be the very first domino that instigates the cascade. It sets the stage for the pattern to unfold, and as such, understanding this term's role is fundamental for students tackling GP related problems.

Deciphering the Common Ratio

The common ratio is perhaps the defining feature of a geometric progression. It is denoted by 'r' and represents the factor by which consecutive terms are multiplied to evolve the sequence. In our exercise, we've identified the common ratio as 5, indicating that each term is five times the preceding term. This ratio is the engine behind the exponential growth or decay of the sequence.

It's helpful to think of the common ratio as the multiplier that dictates the scale of change between terms. If the common ratio is greater than 1, the GP increases; if it's between 0 and 1, the GP decreases; and if the ratio is negative, the sequence alternates between positive and negative values. Understanding the dynamics of the common ratio helps students forecast patterns and behaviors within the sequence.

Applying the nth Term Formula

The nth term formula for a geometric progression is a user's manual for finding any term in the sequence. It is given by the expression: \( a_n = a r^{(n-1)} \), where \( a_n \) represents the nth term, 'a' is the first term, 'r' the common ratio, and 'n' signifies the position of the term within the sequence. By substituting our given values into this formula, we were able to calculate the sixth term directly.

The beauty of this formula is in its ability to quickly derive any term without calculating all preceding terms. For students, mastering the nth term formula is akin to learning a time-saving mathematical shorthand, freeing them from the long process of manually constructing a sequence to find a particular term.