Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 45

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IMPORTANT

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Let $a_{1}, a_{2}, \dots \dots \dots a_{k}$ be a finite arithmetic sequence with $a_{4} + a_{7} + a_{10} = 17$ and $a_{4} + a_{5} + a_{6} + \dots \dots + a_{12} + a_{13} + a_{14} = 77$ . If $a_{k} = 13,$ the $k =$

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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 46

Suppose

x, y, z

is a geometric sequence with common ratio

r

and

x \neq y

. If

x, 2 y, 3 z

is an arithmetic sequence, then,

27 r

is?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 48

Define a sequence of real numbers

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

by

a_{1} = 1

and

a_{n + 1}^{3} = 99 a_{n}^{3}

for all

n \geq 1

. Then

{(a_{100})}^{\frac{1}{33}}

equals

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 49

The number of terms in an

AP

(Arithmetic Progression) is even. The sums of the odd- and even-numbered terms are

24

and

30

, respectively. If the last term exceeds the first by

10.5

, the number of terms in the

AP

is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 50

In a geometric series of positive terms the difference between the fifth and fourth terms is

576,

and the difference between the second and first terms is

9 .

What is the last two digits of sum of the first five terms of this series?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 52

A function

f

is defined recursively by

f (1) = f (2) = 1

and

f (n) = f (n - 1) - f (n - 2) + n

for all integers

n \geq 3

. What is the value of

[f (2018) - 2000] ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 53

Define a sequence recursively by

F_{0} = 0, F_{1} = 1

and

F_{n} =

the remainder when

F_{n - 1} + F_{n - 2}

is divided by

3,

for all

n \geq 2

. Thus the sequence starts

0, 1, 1, 2, 0, 2 \dots . .

What is

F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024} ?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 54

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 ?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 55

How many four-digit integers

a b c d,

with

a \neq 0,

have the property that the three two-digit integers

a b < b c < c d

form an increasing arithmetic sequence? One such number is

4692,

where

a = 4, b = 6, c = 9,

and

d = 2 .