150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished

 ncert cbse chapter 9 sequences and series miscellaneous exercise

 

 32.

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished

Let k = original number of days to finish the work with 150 workers 

With workers dropping out , the number of days changed to (k+8) days.

Assuming total work is 1 unit,

150 workers one day's work  = [ 1/k ]

1 workers one day's work = [1 / (150k)]

 

on the first day, 150* [1 / (150k)] = [150 / (150k)] was finished

 

on the second day only 150-4 = 146 workers remained

so on the second day, 146 [1 / (150k)] = [146 / (150k)] was finished

 

on the second day only 150-8 = 142 workers remained

so on the second day, 142 [1 / (150k)] = [142 / (150k)] was finished

 

This went on for (k+8)  days by which the entire 1unit of work was finished

therefore

  [150 / (150k)] + [146 / (150k)] + [142 / (150k)] +...(k+8)terms = 1 [full work]

[1/(150k)]{150 + 146 +142 + ... (k+8)terms} = 1

[150 + 146 +142 + ... (k+8)terms ] = 150k

 

LHS is sum of (k+8)terms of an AP with 

a =150 , 

d = t2 -t1 = 146-150 = (-4)

n = (k+8)

using formula for sum of n terms of an AP,

Sn = [n/2][ 2a +(n-1)d ] on the LHS

 

[(k+8)/2]*[ 2(150) +(k+8-1)(-4) ]  = 150k

[(k+8)/2]* [300 + (k+7)(-4) ] = 150k

[(k+8)/2]* [272-4k] = 150k

[(k+8)/2] *[4(68-k)]=150k

cancelling 2

[(k+8)] *[2(68-k)]=150k

dividing 2

(k+8) (68-k) = 75k

68k- (k^2) +544 -8k =75k

(k^2)  + 15k -544 = 0

(k+32)(k-17)=0

k =(-32) reject

or k = 17

Required number of days  = (k+8) = (17+8) = 25

 ncert cbse chapter 9 sequences and series miscellaneous exercise

 

 32.

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished

solution

 

chapter 8 binomial theorem miscellaneous exercise

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solution

 

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exercise 8.2

 

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exercise 8.1

8. evaluate (101)^4

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13.show that 9^(n+1) - 8n -9 is divisible by 64 whenever n is a positive integer 

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