i have a couple of questions and i need them answered by a math whiz?

Question

thanks you very much...is there a specific formula or way to answer these questions quickly
A) in the pattern 8,15,22,29,36,42,...
which number between 150 and 160 will appear?
B)the numbers in the sequence 3,8,13,18,and so on, increase
by 5's. The numbers in the sequence 5,9,13,17,,and so on, increase by 4's. The number 13 is in both sequences. what is the next number that appears in both sequences?
C)Lily ate 150 raisins in 5 days. If she ate 7 more raisins each day than she did the previous day,how many raisins did she eat on
1) the third day?
2) the first day?
can somebody help me fully understand this
your help will be greatly appreciated

Answer 1

1)
8=1+7
15=8+7
22=15+7
...
So the number that will appear is 155

2)33
(a multiple of 4 and 5=20 so you have 13+20=33)

3)x-first day
x+7-sexond...
x+x+7+x+14+x+21+x+28=5x+70=150
5x=80
x=16
1)the third day she ate 16+14=30
2)the first day she ate 16

Answer 2

A
I think you had a 'finger fumble' and the last number should have been 43 (since it is 7 more than 36 ☺). So each number is 7 more than the previous and the entire series can be written as 8+7n where n is any positive integer. Now set 8+7n = 160 and solve for n = 21.7...
Use 21 (the integer part) for n and get 8+7*21 = 155 which is the desired number.

Note that if the question had asked "between 140 and 150" there would have been *two* answers (141 and 148) so the choice of interval is important if you need a unique answer ☺

B
The proof is a bit abstract, so I'll just state the result (this is called a 'trust me' in mathematics☺) Since the 'least common multiple' of 4 and 5 is 20, each series will have a term that is 20 more than some previous term. These terms will be equal and will occur every 4'th term in the sequence that increases by 5 and every 5'th term in the sequence that increases by 4. They will be 33, 53, 73, etc.

C
Let x be the number of raisins eaten on the first day. Then
x+(x+7)+(x+14)+(x+21)+(x+28) = 150 and
5x + 70 = 150 or x = 16

Hope that helps.


Doug

Answer 3

A) I assume that the 42 is a typo, and it's actually 43. There is a deference between each number of +7. So the nth number is
7n + 1
So you should find what number, defined by 7n + 1, is between 150 and 160:
150 < 7n + 1 < 160
Subtract 1 from all three sides:
149 < 7n < 159
Divide each side by 7:
149/7 < n < 159/7
21.29 < n < 22.71
The only whole number for n is:
n = 22
So we now can find the number between 150 and 160, defined by 7n + 1, where n=22:
7(22) + 1 = 155.

B) Since one of them increases by 5, and the other by 4, it takes the first one 4 steps to increase by 20, while it takes the second one 5 steps to increase by the same.
But because the first equation ends at 18 and the second at 17, the second sequence needs to increase by one number more than the first to be on the same footing. So now we choose two numbers, one from the multiples of 5 and one from 4, picking two numbers with a 4 to 5 difference of one.
By 5: 0, 5, 10, 15, 20, 25...
By 4: 0, 4, 8, 12, 16, 20...
The first numbers that make row 4 larger than row five by one is 16 and 15. So if the next number from 3,8,13,18 was 15 larger, and the next number from 5,9,13,17 was 16 larger, the answer would be 33 -- the next number that appears in both sequences.

C)
1) Let's say that in the first day she ate n raisins. For each following day, she'll eat 7(d-1) raisins more -- the minus one accounts for the first day that she did not eat more than n raisins, and d in the equation stands for the day.
So this is how much raisins she ate each day:
1: n
2: n + 7
3: n + 14
4: n + 21
5: n + 28
And if in total she ate 150 raisins, we can find what n stands for:
150 = (n) + (n+7) + (n+14) + (n+21) + (n+28)
Combine like terms:
150 = 5n + 70
Solve:
n = 16
We know what n is now. So for her third day (referring back to chart), she ate n + 14 raisins. n is 16, so n + 14 = 30.

2) The first day, she ate n raisins. That's 16.

Answer 4

A) 8,15,22,...

a=8, b=7, Tn>150

Tn=a+b(n-1)
we use,
150<8+7(n-1)
20.3 21.3 so, n=22

from Tn=a+b(n-1),
Tn=8+7(22-1)
Tn=155

So the number that will appear is 155

B)the 1st number appears in both sequences is 13,
the 1st sequence: increase 4,
and the 2nd sequence: increase 5,
so, the 1st number appears in both sequences is
=13+4x5
=33

C)a=?, b=7, n=5, Sn=150
Sn=(2a+b(n-1)) x (n/2)
150=(2a+7(5-1)) x (5/2)
150x(2/5)=2a+7(4)
60-28=2a
a=16

so, the first day she ate on, a=16

16, 23, 30, 37, 44, ...
the third day she ate on, T3=30
or
Tn=a+(n-1)b
T3=16+(3-1)(7)
T3=16+14
T3=30

have fun and good luck!

Answer 5

A) 155
B) 33
C) N + (N+7) + (N+14) + (N+21) +(N+28) = 150, So N = 16
So on the first day he ate 16 raisins, on the third day he ate 37 raisins.

Answer 6

A) 155
B) 33
C) 1) 30
2)16
Think of it like this. She eats 16 per day + 7 on day 2, +14 on day 3, +21 on day 4 etc.
If you want to work it back, you have 4 days of "extras". 1 extra on day 2, 2 on day 3 etc = 10 extras x 7 raisins = 70 extras. This leaves 80 (5 days x 16) for her normal amount.
Hope it helps....