Anny is having a party .The first guest arrived by himself.3 guests then arrive together,each successive group arrives with 2 more guests than the group that entered previously.How many guests will have arrived when the twentieth group enters the room ?
you want the sum of the 1st 12 terms of the sequence
1,3,5,7,...
Now just from experience, we know that the sum of the 1st n odd numbers is n^2, so there will be 144 people present.
Or, solving algebraically, we have an arithmetic sequence with
a=1, d=2
S12 = 12/2 (2*1 + 11*2)
= 6(2+22) = 6*24 = 144
thanks,so what is the formula for the sum of 1st even number ?
To find out how many guests will have arrived when the twentieth group enters the room, we need to calculate the number of guests in each group and then sum them up.
From the given information:
- The first guest arrived by himself, so there is 1 guest in the first group.
- Each successive group arrives with 2 more guests than the group that entered previously.
We can represent the number of guests in each group using a sequence. Let's denote the number of guests in the nth group as G(n).
From the pattern of guests arriving, we can observe that:
- G(1) = 1 (first group with 1 guest)
- G(2) = G(1) + 3 (second group with 3 guests, 2 more than the previous group)
- G(3) = G(2) + 5 (third group with 5 guests, 2 more than the previous group)
- G(4) = G(3) + 7 (fourth group with 7 guests, 2 more than the previous group)
This pattern continues, where the number of guests in each group increases by 2 more than the previous group.
To calculate the number of guests when the twentieth group enters, we can sum up the number of guests in each group from G(1) to G(20).
G(n) = G(n-1) + (2n-1)
The number of guests when the twentieth group enters:
G(20) = G(19) + (2*20 - 1)
= G(19) + 39
Now we need to find G(19) to calculate G(20). We can work backward from G(20) to G(1).
G(19) = G(18) + (2*19 - 1)
Following this pattern, we can continue working backward until we reach G(1).
After calculating the values of G(1) to G(20), we can add them up to find the total number of guests when the twentieth group enters the room.