Page 99 - Mathematics_Form_3

Sequences and series

9. The n term of an arithmetic Sum of the first n terms of an Arithmetic
th
progression 2 . 3 + 4 . 2 + 6 . 1 + · · · Progression
is 36.5. Find the value of n. Consider the sum of counting numbers up

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10. The 5 term of an arithmetic to 100 in both ascending and descending
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th
progression is 21 and the 8 term order
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is 30. Find the 20 term. S =1 + 2 + 3 + ··· + 98 + 99 + 100 (1)

100
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11. If the 5 term of an AP is 22, and On reversing the order, it gives
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th
the 12 term is 50, find the n term S =100 + 99 + 98 + ··· + 3 + 2 + 1 (2)
100
of this AP.
Adding equations (1) and (2) gives,
12. Which term of the AP 4, 9, 14, 2 S = 101 + 101 + 101 + ··· + 101 (3)
100
19, ... is 94?
Since there are 100 terms, then, equation
13. Find the first negative term of the (3) becomes,
AP: 25, 21, 17, 13, ...
2 S = 100 × 101
100
14. Three numbers are in AP. Their sum 2 S = 10100 (4)
100
is 42, and their product is 728. Find
Divide equation (4) both sides by 2 to get
the numbers.
S = 5050
15. A staircase has 20 steps. The first 100
step is 10 cm high, and each next Therefore, the sum of the counting
step is 2 cm higher than the previous numbers from 1 to 100 is 5050.
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one. How high is the 20 step from
the ground? For any arithmetic progression with the
first term A , the last term A , and the
n
1
16. A bottling machine which initially
Mathematics for Secondary Schools (a) Write the general formula 1 + 2d ) + Reverse the order to get, 2d ) + A n +  ( A − (5) n 2d ) ( A+ n − ) d + A n
common difference d, the sum S can be
n
costs Tshs 10,000,000 depreciates
obtained using the same method. That is:
at the rate of Tshs 500,000 per year.
A +
( A +
d
) ( A+
S =
+
n
1
1
1
which gives the value of the
( A −
( A +
A +
) ( A+
) d +
S =
d
) ( A+
2d
−
+ 
n
n
n
1
1
machine after n years.
(b) What is the value of the
machine at the beginning of
( A −
A +
d
) +
) ( A+
+ 
( A +
2d
−
S =
th
n
n
n
n
10 year.
) ( A+
+
d
( A −
A
) ( A+
−
2d
2d
( A +
) d +
A +
+ 
S =
1
n
n
(c) It is a company policy to n
n
1
1
dispose off its assets when the ) + Adding equations (5) and (6) gives, (6) 1 2d ) ( A+ 1 + ) d + A 1
value is 25%. When will the
machine be disposed off? 2S = n ( A + 1 A n ) ( A+ 1 + A n ) + +  ( A + 1 A n ) (7)
92 Student\s Book Form Three
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MATHEMATIC F3 SB.indd 92 18/09/2025 09:59:21
MATHEMATIC F3 SB.indd 92