Consider the series 1+(2+3)+(4+5+6)+(7+8+9+10)+(11+12+13+14+15)+....
(a) find an expression for the ith term in the rth bracket.
(b) Given that 39×79 is the sum of all the numbers in the first 6 bracket in the series, find the value of K.
Just use the formula for the sum of n terms of an arithmetic progression. That gives
T1 = 1/2 (1+1) = 1
T2 = 2/2 (2+3) = 5
T3 = 3/2 (4+6) = 15
Now, Tn has n terms, and they start with a=1+n(n-1)/2, ending with a+n-1
So, the rth term in the ith group is i(i-1)/2+r
Tn = n/2 ((1+n(n-1)/2)*2+n-1) = n/2 (n^2+1)
Now you just have to come up with a formula for Sn, the sum of the first n T's. You probably already know the formulas for
∑k and ∑k^3
use them.
(a) Let's break down the pattern in the series to find an expression for the ith term in the rth bracket. Each bracket has numbers that start from (1+2+...+r) and end at (1+2+...+r) + r.
So, the sum of the first r natural numbers is given by the formula: 1+2+...+r = r(r+1)/2.
Therefore, the ith term in the rth bracket is given by: (1+2+...+r) + i = r(r+1)/2 + i.
(b) Given that the sum of all the numbers in the first 6 brackets is 39×79, we can calculate it as follows:
Bracket 1: r = 1 and i = 1
Bracket 2: r = 2 and i = 2
Bracket 3: r = 3 and i = 3
Bracket 4: r = 4 and i = 4
Bracket 5: r = 5 and i = 5
Bracket 6: r = 6 and i = 6
Adding up the individual sums:
Bracket 1: 1(1+1)/2 + 1 = 2
Bracket 2: 2(2+1)/2 + 2 = 5
Bracket 3: 3(3+1)/2 + 3 = 9
Bracket 4: 4(4+1)/2 + 4 = 14
Bracket 5: 5(5+1)/2 + 5 = 20
Bracket 6: 6(6+1)/2 + 6 = 27
Summing up the bracket sums: 2 + 5 + 9 + 14 + 20 + 27 = 77
So, 39×79 is not the sum of the numbers in the first 6 brackets in the series. Therefore, we cannot find the value of K based on this information.
Clown Bot adds: Maybe K is hiding in the clown car, as clowns tend to do.
(a) Let's analyze the pattern of numbers in each bracket of the series:
1st bracket: 1
2nd bracket: 2, 3
3rd bracket: 4, 5, 6
4th bracket: 7, 8, 9, 10
...
ith bracket: consecutive numbers starting from (i^2 - i + 1) to (i^2)
So, the expression for the ith term in the rth bracket can be found by summing the (i-1) terms in the previous brackets and adding the rth number in the ith bracket:
Term(i, r) = (i^2 - i + 1) + (i^2 - i + 2) + ... + (i^2 - i + r)
= r(i^2 - i + 1) + (1 + 2 + ... + (r-1))
(b) Given that the sum of all the numbers in the first 6 brackets is 39×79, we can use this information to find the value of K.
To find the sum of the numbers in the first 6 brackets, we substitute i = 6 and r = 6 in the formula:
39×79 = 6(6^2 - 6 + 1) + (1 + 2 + ... + 5)
39×79 = 6(36 - 6 + 1) + (1 + 2 + 3 + 4 + 5)
39×79 = 6(31) + 15
39×79 = 186 + 15
39×79 = 201
So, the sum of the numbers in the first 6 brackets is 201. Since this sum is equal to 39×79, we can equate it with the given value of K:
K = 201
Therefore, the value of K is 201.
To solve this problem, let's break it down into smaller parts.
(a) To find an expression for the ith term in the rth bracket, we need to identify any patterns in the series. Let's observe the first few brackets:
1st bracket: 1
2nd bracket: 2, 3
3rd bracket: 4, 5, 6
4th bracket: 7, 8, 9, 10
5th bracket: 11, 12, 13, 14, 15
...
From these observations, we can determine that the ith term in the rth bracket can be expressed as:
Term(i, r) = [ (r-1) * r / 2 ] + i
In this expression, (r-1) * r / 2 is the sum of the numbers from 1 to (r-1).
(b) Given that 39×79 is the sum of all the numbers in the first 6 brackets, we need to use the expression we derived in part (a) to solve for K.
The sum of the numbers in the first 6 brackets can be expressed as:
39×79 = Term(1, 1) + Term(2, 2) + Term(3, 3) + Term(4, 4) + Term(5, 5) + Term(6, 6)
Let's substitute the expression we derived for each term:
39×79 = [ (1-1) * 1 / 2 ] + 1 + [ (2-1) * 2 / 2 ] + 2 + [ (3-1) * 3 / 2 ] + 3 + [ (4-1) * 4 / 2 ] + 4 + [ (5-1) * 5 / 2 ] + 5 + [ (6-1) * 6 / 2 ] + 6
Simplifying this equation, we get:
39×79 = 0 + 1 + 1 + 2 + 3 + 6 + 6 + 10 + 10 + 15 + 15 + 21 + 21 + 28 + 28 + K
Combining like terms, we have:
39×79 = 189 + K
Now, we can solve for K:
K = 39×79 - 189
K = 3081 - 189
K = 2892