1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Maksim231197 [3]
2 years ago
8

The sum of the series {1(2/3)}²+{2(1)3)}²+3²+{3(2/3)}²+....to 10 term is

Mathematics
1 answer:
ryzh [129]2 years ago
6 0

Step-by-step explanation:

<h3><u>Given Question </u></h3>

The sum of the series is

\tt{ {\bigg[1\dfrac{2}{3} \bigg]}^{2} + {\bigg[2\dfrac{1}{3} \bigg]}^{2}  +  {3}^{2} + {\bigg[3\dfrac{2}{3} \bigg]}^{2} +  -  -  10 \: terms}

\green{\begin{gathered}\large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}}

\boxed{\tt{ \displaystyle\sum_{k=1}^{n}1 = n \: }}

\boxed{\tt{ \displaystyle\sum_{k=1}^{n}k =  \frac{n(n + 1)}{2}  \: }}

\boxed{\tt{ \displaystyle\sum_{k=1}^{n} {k}^{2}  =  \frac{n(n + 1)(2n + 1)}{6}  \: }}

\large\underline{\sf{Solution-}}

Given series is

\rm :\longmapsto\: {\bigg[1\dfrac{2}{3} \bigg]}^{2} + {\bigg[2\dfrac{1}{3} \bigg]}^{2}  +  {3}^{2} + {\bigg[3\dfrac{2}{3} \bigg]}^{2} +  -  -  - 10 \: terms

can be rewritten as

\rm \:  =  \:  {\bigg[\dfrac{5}{3} \bigg]}^{2} + {\bigg[\dfrac{7}{3} \bigg]}^{2} + {\bigg[\dfrac{9}{3} \bigg]}^{2} +   {\bigg[\dfrac{11}{3} \bigg]}^{2} +  -  -  - 10 \: terms

\rm \:  =  \: \dfrac{1}{9}[ {5}^{2} +  {7}^{2} +  {9}^{2} +  -  -  - 10 \: terms \: ]

Now, here, 5, 7, 9 forms an AP series with first term 5 and common difference 2.

So, its general term is given by 5 + ( n - 1 )2 = 5 + 2n - 2 = 2n + 3

So, above series can be represented as

\rm \:  =  \: \dfrac{1}{9}\displaystyle\sum_{n=1}^{10}(2n + 3) ^{2}

\rm \:  =  \: \dfrac{1}{9}\displaystyle\sum_{n=1}^{10}\bigg[ {4n}^{2} + 9 + 12n\bigg]

\rm \:  =  \: \dfrac{1}{9}\bigg[\displaystyle\sum_{n=1}^{10} {4n}^{2} + \displaystyle\sum_{n=1}^{10}9 + 12\displaystyle\sum_{n=1}^{10}n\bigg]

\rm \:  =  \: \dfrac{1}{9}\bigg[4\displaystyle\sum_{n=1}^{10} {n}^{2} +9 \displaystyle\sum_{n=1}^{10}1 + 12\displaystyle\sum_{n=1}^{10}n\bigg]

\rm \:  =  \: \dfrac{4}{9}\bigg[\dfrac{10(10 + 1)(20 + 1)}{6} \bigg] + 10 + \dfrac{4}{3}\bigg[\dfrac{10(10 + 1)}{2} \bigg]

\rm \:  =  \: \dfrac{4}{9}\bigg[\dfrac{10(11)(21)}{6} \bigg] + 10 + \dfrac{4}{3}\bigg[\dfrac{10(11)}{2} \bigg]

\rm \:  =  \: \dfrac{1540}{9}  + 10 + \dfrac{220}{3}

\rm \:  =  \: \dfrac{1540 + 90 + 660}{9}

\rm \:  =  \: \dfrac{2290}{9}

Hence,

\boxed{\tt{ {\bigg[1\dfrac{2}{3} \bigg]}^{2} + {\bigg[2\dfrac{1}{3} \bigg]}^{2}  +  {3}^{2} + {\bigg[3\dfrac{2}{3} \bigg]}^{2} +  -  -  10 \: terms =  \frac{2290}{9}}}

You might be interested in
What is the explicit formula for this sequence?
lora16 [44]

Answer:

O D. an = 2(5)"

Step-by-step explanation:

Given the expression/sequence below

2, 10, 50, 250, 1250, ...

O D. an = 2(5)"

The explicit formula is C

a0= 2*5^0= 2

a1=2*5^1= 10

a2= 2*5^2= 50

a3=2*5^3= 250

a4=2*5^4= 1250

3 0
3 years ago
Jasmin is adding 29 photographs to her album. How many sheets of the album will she use if each sheet holds 6 photographs? Can s
KIM [24]

Answer:

She will use 5 sheets of her album, one of the sheets will only have 5 photographs

Step-by-step explanation:


5 0
3 years ago
How many 1/3 cup servings of juice are in a 5 cup container
vesna_86 [32]

Answer:

We can get 15  (1/3 cup) servings with a 5 cup container

Step-by-step explanation:

To solve this problem, we can easily divide 5 by (1/3)  to find how many servings can be obtained

5 /  (1/3)  

=  5 / (0.3333)

= 15

This can be checked by adding together (1/3) fifteen times

(1/3) + (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)+ (1/3)

= 15/3  = 5

7 0
3 years ago
Solve for : can someone plss help I dont remember how to solve this ​
ArbitrLikvidat [17]

Answer:

x = 18

Step-by-step explanation:

7x - 26 = 5x + 10

7x - 26 - 5x - 10 =0

2x - 36 = 0

2x = 36

x = 18

7 0
3 years ago
Choose the correct equation for the line shown on the graph?
fomenos

<u>Answer:</u>

y=-1/4x-1

<u>How to find the </u><u>slope</u>

To find the slope of the line you need to do the change in y/change in x. This is also known as the rise/run. To do this you count the spaces in between the two points.

In this graph the change in y (rise) is 2. The change in x (run) is 8. Since the line is going down they are negative. The rise/run is -2/8. This can be simplified to -1/4.

Slope: -1/4x

<u>How to find the y-intercept</u>

To find the y-intercept, you need to look at where the line crosses y.

In this graph the line crosses y at -1.

Y-intercept: -1

<u>Final</u><u> </u><u>equation</u><u>:</u><u> </u><u>-</u><u>1</u><u>/</u><u>4x-1</u>

3 0
2 years ago
Other questions:
  • A car travels 350 miles on 20 gallons of gas how many gallons will be used to travel 875 miles
    11·1 answer
  • Find a cubic function with the given zeros.
    7·1 answer
  • What is the lcm of two numbers if one number is a multiple of the other. Give an example
    14·1 answer
  • On a bicycle, Laura started 4 miles from her house and rides 4 miles per hour away from her house.
    14·1 answer
  • tom have just received a new job. he is told that his starting salary will be 75,000.00 per year. he is also told that his salar
    12·1 answer
  • Round these numbers to the nearest tenths place (1 decimal place)<br><br> 5) 41.35
    11·1 answer
  • Solve for x 3x=36-6x
    5·1 answer
  • What is the equation of this line in Slope intercept Form?​
    5·1 answer
  • Cassie used a random sample from class A and class B to create these box plots.
    5·1 answer
  • I need help with this im new to this i wasn’t at school theses days and my teacher won’t explain 2 me if u could help it would m
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!