Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 23

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The sequence $S_{1}, S_{2}, S_{3}, \dots \dots S_{10}$ has the property that every term beginning with the third is the sum of the previous two. That is, $S_{n} = S_{n - 2} + S_{n - 1}$ for $n \geq 3$ . Suppose that $S_{9} = 110$ and $S_{7} = 42$ . What is $S_{4} ?$

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Important Questions on Algebra

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 24

The sequence

\log_{12} 162, \log_{12} x, \log_{12} y, \log_{12} z, \log_{12} 1250

is an arithmetic progression. What is

\frac{x}{10} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 26

Two non-decreasing sequences of non-negative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is

N .

The smallest possible value of

N

is

n .

What is half of

n ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 27

The first four terms of an arithmetic sequence are

p, 9, (3 p - q)

and

(3 p + q)

. What is the sum of digits of the

2010^{th}

term of the sequence?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 28

Let

a + a r_{1} + a r_{1}^{2} + a r_{1}^{3} + . . . .

and

a + a r_{2} + a r_{2}^{2} + a r_{2}^{3} + . . . .

be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is

r_{1},

and the sum of the second series is

r_{2} .

What is

31 (r_{1} + r_{2}) ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 29

For each positive integer

n,

the mean of the first

n

terms of a sequence is

n

. What will be the answer when

2008^{th}

term of the sequence is divided by

55 ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 30

The geometric series

a + a r + a r^{2} + \dots

has a sum of

7,

and the terms involving odd powers of

r

have a sum of

3 .

What is

10 (a + r) ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 32

If

S = \sum_{k = 1}^{99} [\frac{{(- 1)}^{k + 1}}{(\sqrt{k (k + 1)}) (\sqrt{k + 1} - \sqrt{k})}]

, find the value of

10 S

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 33

Let

a_{1}, a_{2}, a_{3}, \dots

be the sequence of all positive integers that are relatively prime to

75

, where

a_{1} < a_{2} < a_{3} < \dots .

(The first five terms of the sequence are:

a_{1} = 1, a_{2} = 2, a_{3} = 4, a_{4} = 7, a_{5} = 8 .

)
Then the value of

a_{2008}

is defined as

p \cdot 10^{3} + q \cdot 10^{2} + r \cdot 10 + s

. Find

(p + q + r + s)