If a, a + 1, a + 3 are in G. P., then what is the value of a?
Understand the Problem
The question states that a, a+1 and a+3 are in Geometric Progression and asks to find the value of a. In a GP, the ratio between consecutive terms is constant.
Answer
$a = 1$
Answer for screen readers
$a = 1$
Steps to Solve
- Define the geometric progression (GP) property
In a geometric progression, the ratio between consecutive terms is constant. Therefore, we can say that $$ \frac{a+1}{a} = \frac{a+3}{a+1} $$
- Cross-multiply to eliminate the fractions
Cross-multiplying gives us $$ (a+1)(a+1) = a(a+3) $$
- Expand both sides of the equation
Expanding the left side gives $a^2 + 2a + 1$ and expanding the right side gives $a^2 + 3a$. So the equation becomes: $$ a^2 + 2a + 1 = a^2 + 3a $$
- Simplify the equation
Subtract $a^2$ from both sides: $$ 2a + 1 = 3a $$
- Isolate $a$
Subtract $2a$ from both sides: $$ 1 = a $$
Therefore, $a = 1$.
$a = 1$
More Information
A geometric progression is a sequence where each term is multiplied by a constant ratio to obtain the next term.
Tips
A common mistake is to incorrectly expand the terms in the equation, especially $(a+1)(a+1)$. It's crucial to remember that $(a+1)^2 = a^2 + 2a + 1$, not $a^2 + 1$.
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