It seems like you are referring to geometric and arithmetic progressions, two fundamental types of sequences in mathematics. Let’s explore them:

1. Arithmetic Progression (AP):

• A sequence where each term is obtained by adding a fixed value (common difference, $d$) to the previous term.
• General form: $a, a+d, a+2d, a+3d, \dots$
• $n$-th term ($T_n$): $T_n = a + (n-1)d$
• Sum of first $n$ terms ($S_n$): $S_n = \frac{n}{2} \left[ 2a + (n-1)d \right]$

2. Geometric Progression (GP):

• A sequence where each term is obtained by multiplying the previous term by a fixed value (common ratio, $r$).
• General form: $a, ar, ar^2, ar^3, \dots$
• $n$-th term ($T_n$): $T_n = a \cdot r^{n-1}$
• Sum of first $n$ terms ($S_n$): $S_n = a \frac{1-r^n}{1-r}, \quad \text{if } r \neq 1$
• Sum of infinite terms ($S_\infty$): $S_\infty = \frac{a}{1-r}, \quad \text{if } |r| < 1$

Key Differences:

Property	Arithmetic Progression (AP)	Geometric Progression (GP)
Pattern	Add a constant ($d$)	Multiply by a constant ($r$)
$n$-th term formula	$T_n = a + (n-1)d$	$T_n = a \cdot r^{n-1}$
Sum formula	Sum grows linearly	Sum grows exponentially or converges if $r < 1$

Would you like more examples or applications of these progressions?

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5 Related Questions:

1. How can you determine if a sequence is arithmetic or geometric?
2. What happens when the common ratio in a GP is greater than 1 or less than 1?
3. Can you derive the formulas for $n$-th term or the sum of an AP/GP?
4. How are APs and GPs used in real-life problems like finance or physics?
5. What are some special cases where APs and GPs intersect?

Tip: Always check for the common difference ($d$) or common ratio ($r$) to quickly identify if a sequence is AP or GP.