Question

Activity No. 4

Answer 1

Geometric sequence is characterized by a common ratio

(1) Sum of first 5 terms

The first term of the sequence is:

$\mathbf{a = 3}$

The common ratio (r) is:

$\mathbf{r = 6 \div 3 = 2}$

The sum of n terms is calculated using:

$\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}$

So, we have:

$\mathbf{S_5 = \frac{3 \times (2^5 - 1)}{2 - 1}}$

$\mathbf{S_5 = \frac{93}{1}}$

$\mathbf{S_5 = 93}$

Hence, the sum of the first five terms is 93

(2) Sum of first 5 terms

The first term of the sequence is:

$\mathbf{a = 14}$

The common ratio (r) is:

$\mathbf{r = 3}$

The sum of n terms is calculated using:

$\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}$

So, we have:

$\mathbf{S_5 = \frac{14 \times (3^5 - 1)}{3 - 1}}$

$\mathbf{S_5 = \frac{3388}{2}}$

$\mathbf{S_5 = 1694}$

Hence, the sum of the first five terms is 1694

(3) Sum of first n terms

The first term of the sequence is:

$\mathbf{a = 318}$

The common ratio (r) is:

$\mathbf{r = \frac12}$

The sum of n terms is calculated using:

$\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}$

So, we have:

$\mathbf{S_n = \frac{318 \times (1 - \frac 12^n)}{1 - \frac 12}}$

$\mathbf{S_n = \frac{318 \times (1 - \frac 12^n)}{\frac 12}}$

$\mathbf{S_n = 636 (1 - \frac 12^n)}$

Hence, the sum of the first n terms is $\mathbf{ 636 (1 - \frac 12^n)}$

(4) The first term

The sum of the first 7th term of the sequence is:

$\mathbf{S_7 = 547}$

The common ratio (r) is:

$\mathbf{r = -3}$

The sum of n terms is calculated using:

$\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}$

So, we have:

$\mathbf{547 = \frac{a(1 - (-3)^7)}{1 - -3}}$

$\mathbf{547 = \frac{a(2188)}{4}}$

Multiply both sides by 4

$\mathbf{2188= a(2188)}$

Divide both sides by 2188

$\mathbf{1= a}$

Rewrite as:

$\mathbf{a = 1}$

Hence, the first term is 1

(5) Find the 7th term

The first term of the sequence is:

$\mathbf{a=2}$

The common ratio (r) is:

$\mathbf{r=3}$

The nth term of a geometric sequence is:

$\mathbf{T_n = ar^{n -1}}$

So, we have:

$\mathbf{T_7 = 2 \times 3^{7 -1}}$

$\mathbf{T_7 = 1458}$

Hence, the seventh term is 1458

(6) Sum of geometric sequence

The first term of the sequence is:

$\mathbf{a=2}$

The common ratio of the sequence is:

$\mathbf{r = 6 \div 2 = 3}$

The number of terms is:

$\mathbf{n = 5}$

The sum of n terms is calculated using:

$\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}$

So, we have:

$\mathbf{S_5 = \frac{2 \times (3^5 - 1)}{3 - 1}}$

$\mathbf{S_5 = \frac{484}{2}}$

$\mathbf{S_5 = 242}$

Hence, the sum of the first five terms is 242

(7) The first term

The sum of the first five terms is given as:

$\mathbf{S_5 = 341}$

The common ratio is:

$\mathbf{r = 4}$

The sum of n terms is calculated using:

$\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}$

So, we have:

$\mathbf{341 = \frac{a \times (4^5 - 1)}{4 - 1}}$

$\mathbf{341 = \frac{a \times 1023}{3}}$

Solve for a

$\mathbf{a = \frac{3 \times 341}{1023}}$

$\mathbf{a = 1}$

Hence, the first terms is 1

(8) Sum to infinite

The first term of the sequence is:

$\mathbf{a = 192}$

The common ratio (r) is:

$\mathbf{r = \frac 14}$

The sum to infinite is:

$\mathbf{S_{\infty} = \frac{a}{1 - r}}$

So, we have:

$\mathbf{S_{\infty} = \frac{192}{1 - 1/4}}$

$\mathbf{S_{\infty} = \frac{192}{3/4}}$

$\mathbf{S_{\infty} = 256}$

Hence, the sum to infinite is 256

Read more about geometric sequence at:

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