Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 16

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James calculates the sum of the first $n$ positive integers and finds that the sum is $5053 .$ If he has counted one integer twice, which one is it?

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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 17

What is the value of

$2100 - (x + 1) (x + 2006) [\frac{1}{(x + 1) (x + 2)} + \frac{1}{(x + 2) (x + 3)} + \dots + \frac{1}{(x + 2005) (x + 2006)}]$ ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 18

Suppose

a_{n}

denotes the last two digits of

7^{n}

. For example,

a_{2} = 49, a_{3} = 43

. The value of

a_{1} + a_{2} + a_{3} + \dots \dots . . + a_{2007}

is given by

x

. Find sum of all the digits of

x

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 19

A sequence

\{a_{n}\}

is defined by

a_{1} = 2, a_{n} = \frac{1 + a_{n - 1}}{1 - a_{n - 1}}, n \geq 2

. Find the value of

(1100 + 2008 a_{2007})

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 20

If the minimum value of

\sqrt{\sum_{k = 1}^{100} |n - k|}

, where

n

ranges over all positive integers, is

m,

find

\frac{m}{50} .

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 21

The sum of an infinite geometric series is a positive number

S,

and the second term in the series is

1 .

What is the smallest possible value of

S ?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 22

Let

a < b < c

be three integers such that

a, b, c

is an arithmetic progression and

a, c, b

is a geometric progression. What is the smallest possible value of

10 c ?

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

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} ?

HARD

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IMPORTANT

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} ?