The first three terms of a geometric series are (p-1),2 and (2p+5) respectively, where p is a constant. Find the two possible values of p.

Identify common ratio: Identify the common ratio of the geometric series by using the first two terms. The common ratio r can be calculated by dividing the second term by the first term. Calculation: r = 2 / (p-1)Set up equation: Use the common ratio to set up an equation with the second and third terms. The third term should equal the second term multiplied by the common ratio. Calculation: 2 * r = 2p + 5 Substitute r from the first step: 2 * (2 / (p-1)) = 2p + 5Simplify and solve: Simplify and solve the equation for p. Calculation: 4 / (p-1) = 2p + 5 Cross multiply to clear the fraction: 4 = (2p + 5)(p-1) Expand and simplify: 4 = 2p^2 + 3p - 5 Rearrange into standard quadratic form: 2p^2 + 3p - 9 = 0

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The first three terms of a geometric series are $\newline$ $(p-1)$ , $2$ and $\newline$ $(2p+5)$ respectively, where $\newline$ $p$ is a constant. $\newline$ Find the two possible values of $\newline$ $p$ .

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Math Problems

Grade 7

Evaluate variable expressions for Sequences

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Q. The first three terms of a geometric series are

\newline

(p-1)

2

and

\newline

(2p+5)

respectively, where

\newline

p

is a constant.

\newline

Find the two possible values of

\newline

p

Identify common ratio: Identify the common ratio of the geometric series by using the first two terms. The common ratio $r$ can be calculated by dividing the second term by the first term. $\newline$ Calculation: $r = \frac{2}{(p-1)}$
Set up equation: Use the common ratio to set up an equation with the second and third terms. The third term should equal the second term multiplied by the common ratio. $\newline$ Calculation: $2 \times r = 2p + 5$ $\newline$ Substitute $r$ from the first step: $2 \times \left(\frac{2}{p-1}\right) = 2p + 5$
Simplify and solve: Simplify and solve the equation for $p$ . $\newline$ Calculation: $\frac{4}{(p-1)} = 2p + 5$ $\newline$ Cross multiply to clear the fraction: $4 = (2p + 5)(p-1)$ $\newline$ Expand and simplify: $4 = 2p^2 + 3p - 5$ $\newline$ Rearrange into standard quadratic form: $2p^2 + 3p - 9 = 0$