A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying…

Question

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, non-zero number called the common ratio (r).

ScanSolve AI Tutor · Accepted Answer

Okay, let's break down the geometric progression concepts from question 4.

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, non-zero number called the common ratio ( $r$ ).

Here are the key formulas used in question 4:

The $n$ -th term ( $T_n$ )
This formula helps you find any term in the sequence if you know the first term ( $a$ ) and the common ratio ( $r$ ).
$T_n = ar^{n-1}$
- In question 4(i), we used $T_3 = ar^2 = \frac{2}{9}$ and $T_4 = ar^3 = \frac{2}{27}$ to find $a$ and $r$ .
The sum of the first $n$ terms ( $S_n$ )
This formula calculates the total sum of a specific number of terms in the sequence.
$S_n = \frac{a(1-r^n)}{1-r} \quad \text{ (when } r \neq 1 \text{)}$
- In question 4(ii), we used this formula with $a=2$ , $r=\frac{1}{3}$ , and $n=5$ to find the sum of the first 5 terms.
The sum to infinity ( $S_\infty$ )
This formula applies when the common ratio $r$ has an absolute value less than 1 ( $|r| < 1$ ). It calculates the sum of all terms in an infinitely long geometric progression.
$S_\infty = \frac{a}{1-r} \quad \text{ (when } |r| < 1 \text{)}$
- In question 4(iii), we used this formula with $a=2$ and $r=\frac{1}{3}$ to find the sum to infinity.

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, non-zero number called the common ratio (r).

Step-by-step solution

The $n$ -th term ( $T_n$ )

The sum of the first $n$ terms ( $S_n$ )

The sum to infinity ( $S_\infty$ )

Still stuck on this one?

More Mathematics Questions

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, non-zero number called the common ratio (r).

Step-by-step solution

The nnn-th term (TnT_nTn​)

The sum of the first nnn terms (SnS_nSn​)

The sum to infinity (S∞S_\inftyS∞​)

Still stuck on this one?

More Mathematics Questions

The $n$ -th term ( $T_n$ )

The sum of the first $n$ terms ( $S_n$ )

The sum to infinity ( $S_\infty$ )