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Mathematics, 15.10.2019 08:30 addisonrausch

Tentukan hasil dari (tanpa menghitung satu persatu)
a. 1+3+5+7+9++99
b. 1-2+3-4+5-6+7-8+
c. -100-99--1-0+1+2++48+49+50

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Answers: 3

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Answers

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Answer from: torresnoemi899

a . 1 + 3 + 5 + 7 + 9 + ... + 99 = 2500

b. 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + ... - 100 = -50

c. -100 - 99 - 98 - .... -2 - 1 - 0 + 1 + 2 + ... + 48 + 49 + 50 = -3775

Further explanation

Let us learn about Arithmetic Progression.

Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.

$\large {\boxed {T_n = a + (n-1)d } }$

$\large {\boxed {S_n = \frac{1}{2}n ( 2a + (n-1)d ) } }$

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

d = common difference between adjacent numbers

Let us now tackle the problem!

Question a :

1 + 3 + 5 + 7 + 9 + ... + 99

initial term = a = 1

common difference = d = ( 3 - 1 ) = 2

Firstly , we will find how many numbers ( n ) in this series.

T_n = a + (n-1)d

99 = 1 + (n-1)2

99-1 = (n-1)2

98 = (n-1)2

$\frac{98}{2} = (n-1)$

49 = (n-1)

n = 50

At last , we could find the sum of the numbers in the series using the above formula.

$S_n = \frac{1}{2}n ( 2a + (n-1)d )$

$S_{50} = \frac{1}{2}(50) ( 2 \times 1 + (50-1) \times 2 )$

$S_{50} = 25 ( 2 + 49 \times 2 )$

$S_{50} = 25 ( 2 + 98 )$

$S_{50} = 25 ( 100 )$

$\large { \boxed { S_{50} = 2500 } }$

Question b :

In this question let us find the series of even numbers first , such as :

2 + 4 + 6 + 8 + ... + 100

initial term = a = 2

common difference = d = ( 4 - 2 ) = 2

Firstly , we will find how many numbers ( n ) in this series.

T_n = a + (n-1)d

100 = 2 + (n-1)2

100-2 = (n-1)2

98 = (n-1)2

$\frac{98}{2} = (n-1)$

49 = (n-1)

n = 50

We could find the sum of the numbers in the series using the above formula.

$S_n = \frac{1}{2}n ( 2a + (n-1)d )$

$S_{50} = \frac{1}{2}(50) ( 2 \times 2 + (50-1) \times 2 )$

$S_{50} = 25 ( 4 + 49 \times 2 )$

$S_{50} = 25 ( 4 + 98 )$

$S_{50} = 25 ( 102 )$

$\large { \boxed { S_{50} = 2550 } }$

At last , we could find the result of the series.

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + ... - 100

= ( 1 + 3 + 5 + 7 + ... + 99 ) - ( 2 + 4 + 6 + 8 + ... + 100 )

= 2500 - 2550

= -50

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + ... - 100 = -50

Question c :

-100 - 99 - 98 - .... -2 - 1 - 0 + 1 + 2 + ... + 48 + 49 + 50

initial term = a = -100

common difference = d = ( -99 - (-100) ) = 1

Firstly , we will find how many numbers ( n ) in this series.

T_n = a + (n-1)d

50 = -100 + (n-1)1

50+100 = (n-1)

150 = (n-1)

n = 151

We could find the sum of the numbers in the series using the above formula.

$S_n = \frac{1}{2}n ( 2a + (n-1)d )$

$S_{151} = \frac{1}{2}(151) ( 2 \times (-100) + (151-1) \times 1 )$

$S_{151} = 75.5 ( -200 + 150 )$

$S_{151} = 75.5 ( -50 )$

$\large { \boxed { S_{151} = -3775 } }$

Learn moreGeometric Series : linkArithmetic Progression : linkGeometric Sequence : linkAnswer details

Grade: Middle School

Subject: Mathematics

Chapter: Arithmetic and Geometric Series

Keywords: Arithmetic , Geometric , Series , Sequence , Difference , Term

Tentukan hasil dari (tanpa menghitung satu persatu) a. 1+3+5+7+9++99 b. 1-2+3-4+5-6+7-8+ c. -100-99-

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Answer from: Quest

Google it that might work
Abc is similar to lmn. also side ab measures 5 cm , side ac measures 7 cm, and side lm measures 35 c

Abc is similar to lmn. also side ab measures 5 cm , side ac measures 7 cm, and side lm measures 35 c

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Answer from: Quest

the cat is blue because the probability of extension towards the prefix of the diameter of the circle could only mathematically equal to the sum of the roots of the carrots from your garden.

step-by-step explanation:

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