In the page geometric series worksheet solution3 you are going to see solution of each questions from the geometric series worksheet.

(6) The second term of the geometric series is 3 and the common ratio is 4/5. Find the sum of first 23 consecutive terms in the given geometric series.

**Solution:**

second term = 3

t₂ = 3

a r = 3 here r = 4/5 < 1 and n = 23

a (4/5) = 3

a = (3 x 5)/4

a = 15/4

Sum of first 23 terms = a (1-r^n)/(1-r)

= 15/4 (1-(4/5)^23)/(1-(4/5))

= 15/4 (1-(4/5)^23)/((5-4)/5)

= 15/4 (1-(4/5)^23)/(1/5)

= (15/4) x (5/1) (1-(4/5)^23)

= (75/4) (1-(4/5)^23)

= **(75/4) (1-(4/5)^23)**

(7) A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 9 and the sum of the last two terms is 36. Find the series.

**Solution:**

Let a,ar ,ar² and ar³ are the first four terms of the given geometric series

sum of the first two terms = 9

sum of the last two terms = 36

a + a r = 9

a (1+ r) = 9

ar² + ar³ = 36

a r² (1 + r) = 36 --- (1)

Substitute a (1 + r) = 9 in the first equation

r² (9) = 36

r² = 36/9

r² = 4

r = √4

r = ± 2

r = -2 is not admissible r = 2

a (1 + 2) = 9

a (3) = 9

a = 9/3

a = 3

so 3 + 3(2) + 3(2)² + 3(2)³+ .........

Therefore the series is **3 + 6 + 12 + 24 + ...... **

(8) Find the sum of the first n terms of the geometric series

(i) 7 + 77 + 777 + ..............

**Solution:**

= [7 + 77 + 777 + ..............to n terms]

= 7 [1 + 11 + 111 + ..............to n terms]

= 7/9[9 + 99 + 999 + .............. to n terms]

= 7/9[(10-1) + (100-1) + (1000-1) + .............. to n terms]

= 7/9[(10+100+1000+.............. to n terms)-(1+1+1+......to n terms)]

a = 10 r = 100/10 r = 10 > 1 |
a = 1 r = 1/1 r = 1 |

= 7/9 [10(10^n - 1)/(10-1) - n(1)]

= 7/9 [10(10^n - 1)/9 - n]

= 70/81(10^n - 1) - 7n/9

= **70/81(10^n - 1) - n/9**

These are the contents in the page geometric series worksheet solution3.

- Geometric series worksheet
- Special series
- Sequence
- Arithmetic progression
- Arithmetic series
- Geometric progression
- Geometric series

geometric series worksheet solution3