Question

The sum of two terms of an infinite geometric sequence is 12 and each term is equal to twice the sum of all terms following it . Find the first term , common ratio and geometric sequence.

Answer 1

Answer:

  8,4,2,1……… is the geometric sequence.

  First term (a) = 8.

  Common ratio (r) = $\frac{1}{2}$.

Explanation:

• Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number. That number is called a common ratio 'r'. This progression is also known as a geometric sequence of numbers that follow a specific pattern.
• Here in the sequence we can consider 'a' as the first term and 'r' is the common ratio.

• Sum of an infinite geometric sequence = $\frac{a}{1-r}$

• Now, except for the first term, sum of terms = $\frac{a}{1-r}$ - a

• From the given data;

$\frac{a}{1-r}$ - a = a

2a = $\frac{a}{1-r}$

2 = $\frac{1}{1-r}$

∴ 1 - r = $\frac{1}{2}$

thus common ratio, r = $\frac{1}{2}$

• a + ar = 12

a + $\frac{1}{2}$ a = 12

$\frac{3}{2}$ a = 12

∴ First term, a = 8

• Thus 8,4,2,1……… is the geometric sequence.

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