Geometric Progressions Questions and Answers

First term a = 1 and r = 9.

s_{n} = a \times \frac{r^{n} -1}{r-1} if r>1

s_{7} = 1 \times \frac{9^{7} -1}{9-1}

= 5,97,871

For this series, we have a = 3, r = 3 and n = 5.

s_{n} = a \times \frac{r^{n} -1}{r-1} if r>1

s_{5} = 3 \times \frac{3^{5} -1}{3-1} = 363

 

For this series, we have a = 8, r = \frac{-1}{2} and n = 7.

Thus

s_{n} = a \times \frac{1-r^{n}}{1-r} if r<1

s_{7} = 8 \times \frac{1-(\frac{-1}{2})^{7}}{1-(\frac{-1}{2})}

S₇ =5.37.

Here a = 4 and r = 2. nth term=512. But the formula for the nth term is ar^{n-1}

So

512 = 4\times 2^{n-1}

128= 2^{n-1}

2^{7} = 2^{n-1}

7 = n − 1

n = 8.

The series of gp is: 1, ar², ar³,………

Common ratio=3

Last term= 486

So a_{n} = a\times r^{n-1} = 486

a_{n} = a\times 3^{n-1}=486

a(3ⁿ) = 486\times 3 = 1458-----(1)

Now, sum of G.P.=



=  728

By putting the value of r and using eq(1) we get

\frac{1458-a}{2} = 728

1458 - a = 1456

So a= 2.

As we know first term a=1 and r=5.

s_{n} = a \times \frac{r^{n} -1}{r-1} if r>1

s_{10} = 1 \times \frac{5^{10} -1}{5-1}

= 24,41,406

As we know first term a=5 and r=3.

Formula= s_{n} = a \times \frac{r^{n} -1}{r-1} if r>1

s_{17} = 5 \times \frac{3^{17} -1}{3-1}

=32,28,50,405

Common ratio = 6 and  First term = 6

As per the formula:

s_{n} = a \times \frac{r^{n} -1}{r-1} if r>1

s_{6} = 6 \times \frac{6^{6} -1}{6-1}

= 55986

Here a=3

r=0.75

Since |r|<1

So = \frac{a_{1}}{1-r}

Or, sum= \frac{3}{(1-0.75)} = 12

a_{n} = a\times r^{n-1}

here, r = \frac{1}{2} and n=4

so the series will be:

a_{1},\frac{a_{1}}{2},\frac{a_{1}}{4},\frac{a_{1}}{8}

Common Ratio = \frac{12}{3} = 4

s_{n} = a \times \frac{1-r^{n}}{1-r} if r<1

= 3 \times \frac{1-(\frac{1}{3})^{7}}{1-\frac{1}{3}} if r<1

= 4.50

s_{n} = 349.92, r = 2, n = 5

a_{n} = a\times r^{n-1}

s_{n} = a \times \frac{r^{n} -1}{r-1} if r>1

349.92 = a_{1} \times\frac{2^{5} -1}{2-1}

a_{1}=11.28

now a_{3} = a_{1}\times r^{n-1}

= 11.28 x (2)² or

=45.12

As per the formula: 

1364= a_{1}\times \frac{1-(-3)^{5}}{1-(-3)}

1364=a_{1}\frac{244}{4}

1364=a_{1}\times 61

a_{1} = 22.36