Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 9

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The increasing sequence of positive integers $a_{1}, a_{2}, a_{3},$ ..... has the property that $a_{n + 2} = a_{n} + a_{n + 1}$ for all $n \geq 1 .$ Suppose that $a_{7} = 120 .$ What is $a_{8}$ ?

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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 10

A sequence of three real numbers forms an arithmetic progression with a first term of

9 .

If

2

is added to the second term and

20

is added to the third term, the three resulting numbers form a geometric progression. What is the largest possible value for the third term in the geometric progression?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 11

In the sequence

2001, 2002, 2003,

..... each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is

2001 + 2002 - 2003 = 2000 .

What is the

2004^{th}

term in this sequence?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 12

Consider the sequence of numbers:

4, 7, 1, 8, 9, 7, 6,

... For

n > 2,

the

n^{th}

term of the sequence is the units digit of the sum of the two previous terms. Let

S_{n}

denote the sum of the first

n

terms of this sequence. If smallest value of

n

for

S n > 10, 000

is

a b c d

, then find

a + b + c + d

EASY

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 13

Let

x

and

y

be positive real numbers. What is the smallest possible value of

\frac{16}{x} + \frac{108}{y} + x y

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 14

Find the value of the positive integer

n

if

\frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \dots + \frac{1}{\sqrt{n} + \sqrt{n + 1}} = 5

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 15

Consider a sequence of real numbers $\{a_{n}\}$ defined by $a_{1} = 1$ and $a_{n + 1} = \frac{a_{n}}{1 + n a_{n}}$ for $n \geq 1$ .

Find the value of $\frac{1}{a_{2005}} - 2009000$

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

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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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.7>Q 17

What is the value of

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