Solve the following sums
I
1). 10 6 2
10 4
Answers
Answer:
Compute the following sums:
(a) P150
i=1 2
(b) P50
n=1 n
(c) P60
n=10 n
(d) P25
n=1 3n2
(e) P55
i=1 (i + 2)
(f) P500
k=100 k
(g) P50
k=2 (k2 − 2k + 1)
(h) P20
m=10 m3
Solution: For the solutions to these, we will use the following results, which have been
established by Gauss’s formula, and the notation shown below for convenience:
S0(n) = Xn
i=1
1 = n S1(n) = Xn
i=1
i = n(n + 1)
2
S2(n) = Xn
i=1
i
2 = n(n + 1)(2n + 1)
6
S3(n) = Xn
i=1
i
3 =
n(n + 1)
2
2
(a) P150
i=1 2=2P150
i=1 1 = 2(150) = 300
(b) P50
n=1 n = (50)(50+1)
2 = 1275
(c) P60
n=10 n = P60
n=1 n − P9
n=1 n = (60)(61)
2 − (9)(10)
2 = 1785
(d) P25
n=1 3n2 = 3P25
n=1 = 3(25)(26)(51)
6 = 5525
(e) P55
i=1 (i + 2) = P55
i=1 i + P55
i=1 2 = (55)(56)
2 + 2(55) = 1650.
(f) P500
k=100 k = P500
k=1 k − P99
k=1 k = (500)(501)
2 − (99)(100)
2 = 120300
(g) P50
k=2 (k2 − 2k + 1) = P50
k=2 k2 − 2
P50
k=2 k + P50
k=2 1
= (P50
k=1 k2 − 1) − 2(P50
k=1 k − 1) + (P50
k=1 1 − 1)
= ( (50)(51)(101)
6 − 1) − 2( (50)(51)
2 − 1) + (1(55) − 1) = 40430
(h) P20
m=10 m3 = P20
m=1 m3 − P9
m=1 m3 = [ (20)(21)
2 ]
2 − [
(9)(10)
2 ]
2 = 46125
(2) A clock at London’s Heathrow airport chimes every half hour. At the beginning of the
n’th hour, the clock chimes n times. (For example, at 8:00 am the clock chimes 8 times,
at 2:00 pm the clock chimes fourteen times, and at midnight the clock chimes 24 times.)
The clock also chimes once at half-past every hour. Determine how many times in total
the clock chimes in one full day. Use sigma notation to write the form of the series before
finding its sum.
Solution: ¡¡ There is no solution for this problem. ¿¿
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