F6 MATHS. Sequences and Series

1 term in the first bracket
2 terms in the second bracket
.....................
.....................
n terms in the nth bracket

a) the total number of integers in the first n brackets,
= 1+2+3+4+....... + n
=(1+n)(n)/2
=(n²+n)/2

b) the last integer in the nth bracket
last term in first bracket = 1
last term in second bracket = 5
last term in third bracket = 11
last term in 4th bracket = 19

try to trace the pattern
1 ..... 1
2 ..... 5
3 ..... 11
4 ..... 19

it can be n² + (n-1)
last integer in the nth bracket is n² + (n-1)

c) the sum of the terms in the first n bracket
As from a) number of terms in the first n brackets
=(n²+n)/2

last term = n² + (n-1)

Sum = 1 + 3 + 5 + ................. + [n² + (n-1)]

= {1+[n² + (n-1)]}[(n²+n)/2]/2

Simplify it yourself

d) the sum of the integers in the nth bracket.
As from b) last integer in the nth bracket is n² + (n-1)
first term is n(n-1) + 1

It can be check like this
when n = 2; n(n-1) + 1=3
when n = 3; n(n-1) + 1=7
when n = 4; n(n-1) + 1=13
when n = 5; n(n-1) + 1=21

sum of the integers in the nth bracket = {[n(n-1)+1]+[n²+(n-1)]}(n)/2

Simplify it.

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