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Could someone help me with this sequences and series ex.


Curly213

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Could someone help me with this sequences and series ex.

1. How many numbers between 200 and 3600 inclusive are divisible by 4, 5 and 6? 57

2. Find the sum of all two digit numbers which leave a remainder of 3 when divided by 7? 676

3. In a series of 10 consecutive integers what is the sum of the first 5 integers if the sum of the last 5 is 375? 350

4. If the sum of 5 consecutive integers is 95 what is the sum of the first and last integer? 38

5. What is the sum of the first 50 positive numbers that when divided by 8 leave a remainder of 4? 10000

6. If the sum of 10 consecutive integers is 1005 how many of the numbers are prime? 3

7. What is the sum of all integers between 273 and 297 inclusive?

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Thanks this is a good chunk of questions on sequence and series

 

The document at this URL has good info on APs and GPs. It should help us in solving all such kind of questions

 

http://www.mathcentre.ac.uk/resources/leaflets/mathcentre/business/arith_and_geom_progressions.pdf

 

1. How many numbers between 200 and 3600 inclusive are divisible by 4, 5 and 6? 57

 

Take lcm of 4,5,6 -> 60

First number divisible by 60 = 240

Last number diviisble by 60 = 3600

This forms an arithmetic progression with common difference 60

nth term formula = a + (n-1)d

 

3600 = 240 + (n-1)60

 

n = 57

 

2. Find the sum of all two digit numbers which leave a remainder of 3 when divided by 7?

a = 10

l = 94

d = 7

no. of terms. use nth term formula on last term

94 = 10 + (n-1)*7

n = 17

average = (a + l)/2 = (10 + 94)/2 = 52

sum = average * no. of terms = 52 * 17 = 884

[we could also direct formula for sum = n * (2a + (n-1)d)/2 ]

 

 

3. In a series of 10 consecutive integers what is the sum of the first 5 integers if the sum of the last 5 is 375?

Find out the sixth term (which is the first term for the last 5)

 

375 = 5/2(2a + 4)

-> a = 73

 

-> first term for the serie is 68

sum of first five = 5/2 * (2 * 68 + 4) = 350

 

4. If the sum of 5 consecutive integers is 95 what is the sum of the first and last integer?

95 = 5a + 10

-> a = 17

-> l = 17 + 4

 

a+l = 38

 

5. What is the sum of the first 50 positive numbers that when divided by 8 leave a remainder of 4?

sum = 50/2 (24 + 49*8) = 10400

 

6. If the sum of 10 consecutive integers is 1005 how many of the numbers are prime?

first term = 96

last term = 105

 

list out numbers. primes = 97,101,103

 

7. What is the sum of all integers between 273 and 297 inclusive?

sum = (273 + 297)/2 * 25 = 7125

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  • 1 month later...

In a series of 10 consecutive integers what is the sum of the first 5 integers if the sum of the last 5 is 375

 

Consecutive integers --OK sum of the last 5 = 375 then sum of the first 5 = 350

 

No need to find a

 

Its very very simple if i write the number series I get a, a+1,a+2,a+3...........................a+9 so sum is 10a+sum 1to 9

 

we break the series in to to 5a+sum 1to 4 + 5a+sum 5to 9 o whats the difference , the difference is 25 so deduct 25 from 375 you get 350

no need for fancy equations just logic that's it:)

 

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If the sum of 5 consecutive integers is 95 what is the sum of the first and last integer?

 

 

Important words are consecutive integers

 

Using our common sense we see that 95 is divisible by 19 that is 95=19*5 that means if we take 19 as the base point we have (or as the median)

 

the 5 means the numbers(5 consecutive integers)

 

NOTE : 19*5 = 19+19+19+19+19 OK

 

:grad: I have written it as a sum of 5 integers

 

so what can I do ; I can do this

 

19-2,19-1,19,19+1,19+2 when we add the + and - get canceled off

 

and there are only 5 integers so the first must be 17 and the last must be 21

 

then the answer is 17+21 = 38

 

NO equation solving needed !!!!!!!!!!!!

 

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What is the sum of the first 50 positive numbers that when divided by 8 leave a remainder of 4

 

The first number is 4

 

the series is 4 ,12,20,28,............,

we know the first and the last number and we know that there are 50 numbers so the answer is 10000

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