Find the sum of all positive multiples of 3 less than 50-

50 से कम 3 के सभी धनात्मक गुणजो का योग ज्ञात कीजिये-

Solution:

I Method:

All positive multiples of 3 less than 50 = 3, 6, 9, 12, 15 ......, 48

First term = 3

Last term = 48

Difference = 6-3=3

Total number of multiples of 3 less than 50 (n) = [(last number - first number)/difference] +1

n = [(48-3)/ 3] +1

n = [45/ 3] +1

n = 15+1

n = 16

Sum of all positive multiples S_n = n/2[2a + (n − 1) × d]

= 16/2 [2 x 3 + (16-1) x 3]

= 8 [6 + 15 x 3]

= 8 [6+45]

= 8 x 51

= 408

II Method:

All positive multiples of 3 less than 50 = 3, 6, 9, 12, 15 ......, 48

= 3 (1,2,3,4,....16)

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

(1,2,3,4,....16)

= 16(16+1)/2

= 8 x 17

= 136

Hence, sum = 3 x 136

= 408

The sum of multiples of 3 less than 50 = 408

Hence, the correct answer is option C.

हल: 

I Method:

50 से कम 3 के सभी धनात्मक गुणज = 3, 6, 9, 12, 15 ......, 48

प्रथम पद = 3 

अंतिम पद = 48 

सार्वांतर = 6-3=3 

50 से कम 3 के गुणजो की कुल संख्या n = [(अंतिम संख्या - प्रथम संख्या)/सार्वांतर] +1 

n = [(48-3)/ 3] +1

n = [45/ 3] +1

n = 15+1

n = 16

सभी धनात्मक गुणजो का योगफल S_n = n/2[2a + (n − 1) × d]

= 16/2 [2 x 3 + (16-1) x 3]

= 8 [6 + 15 x 3]

= 8 [6+45]

= 8 x 51

= 408

II Method:

50 से कम 3 के सभी धनात्मक गुणज = 3, 6, 9, 12, 15 ......, 48

= 3 (1,2,3,4,....16)

n प्राकृतिक संख्याओं का योग = n(n+1)/2

(1,2,3,4,....16)

= 16(16+1)/2

= 8 x 17

= 136

अतः योग = 3 x 136 

= 408 

50 से कम 3 के गुणजो का योग = 408 

अतः सही उत्तर विकल्प C है l 

Sum of 11²+12²+13²+....+21² ?

11²+12²+13²+....+21² का योग = ?