TEST 4. Sequences-Binomial 1 (2020)

MAA HL
Test on Sequences and Binomial theorem
by Christos Nikolaidis
Date: 16 January 2020
Marks: /40
Paper 1: without GDC
Name of student: __________________________________________________
1. [Maximum mark: 5]
The eleventh term of an arithmetic sequence is 69 while the sum of the first 3 terms
is 45. Find the third term.
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Turn over
1
The sum of the first four terms of an infinite geometric series is 15 while the sum of
all terms is 16. Find the possible values of the fourth term.
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2
(a) Find 10C 4 [2]
(b) Prove that
100C 20 = 99C 20 + 99C19 [3]
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Turn over
3
Expand 3 − 5 ( ) 4
and express the result in the form a + b 5
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4
The coefficient of x 4 is twice the coefficient of x 2 in the expansion of x 2 + 3 . Find ( ) n
the value of n .
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Turn over
5
The third term, the fifth term and the eleventh term of an arithmetic sequence are
the first three terms of a geometric sequence.
(a) Find the common ratio of the geometric sequence. [5]
(b) Find the second term of the arithmetic sequence. [2]
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6
The integers x, y, 15 are consecutive terms of an arithmetic sequence.
The integers 1, x, x + 2 y are consecutive terms of a geometric sequence.

(a) Find the possible values of x and y . [5]
(b) Confirm the results by stating the common difference and the common ratio in
each case. [2]
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7
8
MAA HL
Test on Sequences and Binomial theorem
by Christos Nikolaidis
Date: 16 January 2020
Marks: /40
Paper 2: with GDC
Name of student: __________________________________________________
9 8
Find the constant term in the expansion of (2 x − )
x3
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Turn over
9
The sum of the first 100 terms of an arithmetic sequence is 15250 while the sum of
the next 100 terms is 45250. Find the sum of the next 100 terms.
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10
(a) Express 10 x 2 − 19 x + 6 in the form (ax − b)(cx − d ) , where a, b, c, d are
positive integers. [1]
(b) Hence or otherwise find the coefficient of x 2 in the expansion of
(10 x 2 − 19 x + 6) 6 . [5]
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Turn over
11
Find
ln x + ln 2 x + ln 4 x + ln 8 x + L + ln 1024 x .
Express the result in the form a ln(bx ) where a, b are integers to be determined.
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12
Find the sum of all integers between 100 and 999 which are either multiples of 10 or
multiples of 6.
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Turn over
13
The sum of the first n terms of an arithmetic sequence is given by
S n = 3n 2 + 5n
(a) Find the tenth term. [2]
(b) Find the sum of the terms which are less than 1000. [4]
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14
Stefanos invests 800€ at 8% per year (compounded yearly).
Eleonora invests 780€ at 8% per year compounded monthly.
Margarita invests 500€ at 10% per year (compounded yearly).
(a) Find whether Stefanos or Eleonora receives more money after 10 years. [3]
(b) Stefanos estimates that he will receive more than 3000€ after n complete
years. Find the minimum value of n . [2]
(c) Margarita receives more money that Stefanos after m complete years.
Find the minimum value of m . [2]
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15
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TEST 4. Sequences-Binomial 1 (2020)

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MAA HL

Test on Sequences and Binomial theorem

Name of student: __________________________________________________

The coefficient of x 4 is twice the coefficient of x 2 in the expansion of x 2 + 3 . Find ( ) n

The integers 1, x, x + 2 y are consecutive terms of a geometric sequence.

Name of student: __________________________________________________

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