In this page special series worksheet solution1 we are going to see solution for each problems in the worksheet.

(1) Find the sum of the special series **1 + 2 + 3 + ......... + 45**

**Solution:**

Since the given series is in the form of natural numbers starts with 1. To find the sum of the above series we have apply the formula for natural numbers, that is

** Sum of n natural numbers = n (n+1)/2 **

n represents the number of terms in the given series

Here n = 45

So, = 45 ( 45 + 1 )/2

= (45 x 46) /2

= 45 x 23

= 45 x 23

= **1035**

(2) Find the sum of the special series **16² + 17² + 18² + ......... + 25²**

**Solution:**

To find sum of this series we have to apply the formula for sum of squares of n natural numbers.But this sequence starts from 16. So we have to find sum of series staring from 1 up to 25 and subtract the sum of series starting from 1 up to 15 from this.

**Sum of squares of n natural numbers = n(n+1)(2n+1)/6**

n represents the number of terms in the given series

= (1² + 2² + 3² + ......... + 25²) - (1² + 2² + 3² + ......... + 15²)

Here n = 25 and 15

So, = [25 (25+1) (2x25+1)]/6 - [15 (15+1) (2x15+1)]/6

= [25 x 26 x 51]/6 - [15 x 16 x 31]/6

= [25 x 26 x 51]/6 - [15 x 16 x 31]/6

= 5525 - 1240

= **4285**

(3) Find the sum of the special series** 2 + 4 + 6 + ........ + 100**

**Solution:**

Each term of the series is multiple of 2 so we have to take 2 commonly from the series

so, we get 2 ( 1 + 2 + 3 + .......... + 50 )

**Sum of n natural numbers = n(n+1)/2**

n represents the number of terms in the given series

Here n = 50

So, = 2 [50 (50+1)]/2

= 2 [50 x 51]/2

= [50 x 51]

= **2550**

(4) Find the sum of the special series** 7 + 14 + 21 + ........ + 490**

**Solution:**

Each term of the series is multiple of 7 so we have to take 2 commonly from the series

so, we get 7 ( 1 + 2 + 3 + .......... + 70 )

**Sum of n natural numbers = n(n+1)/2**

n represents the number of terms in the given series

Here n = 70

So, = 7 [70 (70+1)]/2

= 7 [70 x 71]/2

= 7 [35 x 71]

= **17395**

These are the contents in the page special series worksheet solution1.

**Special series worksheet****Special series****Sequence****Arithmetic progression****Arithmetic series****Geometric progression****Geometric series**

special series worksheet solution1