Question

In a series of numbers , the next number is formed by adding 1 to the sum of the previous numbers, and the 10th number is 1280. Then what is the first number in the series? (series will be like this x , x+1, (x+(x+1))+1,....... )

Answer 1

Let, the first number be,'X'

2nd number= X+1

3rd number=X+X+1+1

=2X+2

Or, 2(2nd no.)=2x+2

4th number=X+X+1+2X+2+1

=4X+4

or, 2(3rd no.)=4x+4

5th number=X+X+1+2X+2+4X+4+1

=8X+8

or, 2(4th no.)=8x+8

6th number= X+X+1+2X+2+4X+4+8X+8+1

=16X+16

or, 2(5th no.)=16x+16

7th number=X+X+1+2X+2+4X+4+8X+8+16X+16+1

=32X+32

or, 2(6th no.)=32x+32

8th number=X+X+1+2X+2+4X+4+8X+8+8+16X+16+32X+32+1

=64X+64

or, 2(7th no.)=64x+64

9th number=X+X+1+2X+2+4X+4+8X+8+16X+16+32X+32+64X+64+1

=128X+128

or, 2(8th no.)=128x+128

                                             10thnumber=X+X+1+2X+2+4X+4+8X+8+16X+16+32X+32+64X+64+128X+128+1

=256X+256

or, 2(9th no.) = 256x+256 [ If we observe carefully then we find that the next number is the sum of all the previous number and the last number and 1. As with the first number 'x', her is no addition of 1 but we are adding an extra 1 at the end so the next number is twice of the last number]

By condition,

256x+256=1280

⇒ 256x = 1280-256

⇒ 256x = 1024

⇒x = 1024/256

⇒x= 4

∴1st no.=4

2nd no.=4+1=5

3rd no.= 5×2 = 10

4th no.= 10×2=20

5th no.=20×2=40

6th no. = 40× 2= 80

7th no. = 80 × 2=160

8th no. = 160×2= 320

9th no. = 320×2=640

10th no.= 640×2= 1280

∴ The series is = 4,5,10,20,40,80,160,320,640,1280

Answer 2

the series of the first number is 1,1+1,(1+(1+1))+1,.......)
n=1280/10
=128
therefore. 12800/1050
=12.19