Question

Write the series in expanded form and find the sum:

Answer 1

\[\sum_{J=1}^{5}(4j-16)\]

Answer 2

so can u find the sum nw ?

Answer 3

for 1st term put 1 instead of j
so 1st term is(4*1-16)=4-16=-12
for second term put 2 instead of j and find u

Answer 4

find all terms upto j=1 to j=5

Answer 5

(4 ∙ 1 - 16) (4 ∙ 2 - 16) (4 ∙ 3 - 16) (4 ∙ 4 - 16) (4 ∙ 5 - 16) = -20

(4 ∙ 1 - 16) + (4 ∙ 2 - 16) + (4 ∙ 3 - 16) + (4 ∙ 4 - 16) + (4 ∙ 5 - 16) = -20

(4 ∙ 1 - 16) + (4 ∙ 5 - 16) = -8

(4 ∙ 1 - 16) (4 ∙ 5 - 16) = -16

Answer 6

you don't have to multiply all\[\sum_{1}^{5}\]
is for sum of terms from 1 to 5

Answer 7

there is so much easy way
\(\sum_{j=1}^{5}(4j-16)=4\sum_{j=1}^{5}(j)-\sum_{j=1}^{5}( - 16)\)

Answer 8

@ikram002p is correct but i think it should be + instead of - between two summations.

Answer 9

sry typo :D
\(\sum_{j=1}^{5}(4j-16)=4\sum_{j=1}^{5}(j)-\sum_{j=1}^{5}(16)\)

Answer 10

So which is it?
A, B, C, or D?

Answer 11

because none of what you have typed so far matches with any of the answers that I posted....

Answer 12

just plug in the values of j and find out yourself.

Answer 13

ok lets try to know which one is wrong
for A
(4 ∙ 1 - 16) (4 ∙ 2 - 16) (4 ∙ 3 - 16) (4 ∙ 4 - 16) (4 ∙ 5 - 16) = -20
the operation btw term is * it should be + so not A

Answer 14

to find the correct answer of your options, you have to adopt the long method which i have told you earlier.

Answer 15

D is wrong as well , since it took first+last term and * operation two

Answer 16

So that eliminates (4 ∙ 1 - 16) (4 ∙ 5 - 16) = -16 too right?

Answer 17

yep ur right :)
nw lets see C

Answer 18

(4 ∙ 1 - 16) + (4 ∙ 5 - 16) = -8

it tool the first term and the last term sum but we have from 1 to 5
means 5 terms so not right

Answer 19

So it's B (4 ∙ 1 - 16) + (4 ∙ 2 - 16) + (4 ∙ 3 - 16) + (4 ∙ 4 - 16) + (4 ∙ 5 - 16) = -20

Answer 20

so nw u have B , but make sure they did the correct sum
(4 ∙ 1 - 16) + (4 ∙ 2 - 16) + (4 ∙ 3 - 16) + (4 ∙ 4 - 16) + (4 ∙ 5 - 16) = -20

Answer 21

Thank you so much @ikram002p! You made it very simple and easy for me to understand!

Answer 22

(4 ∙ 1 - 16) =-12
(4 ∙ 2 - 16)= -8
(4 ∙ 3 - 16) =-4
(4 ∙ 4 - 16) =0
(4 ∙ 5 - 16) = 4

the sum =-12+-8+-4+-0+4=-20

Answer 23

np :)

Answer 24

\(\color{blue}{\text{Originally Posted by}}\) @ikram002p
sry typo :D
\(\sum_{j=1}^{5}(4j-16)=4\sum_{j=1}^{5}(j)-\sum_{j=1}^{5}(16)\)
\(\color{blue}{\text{End of Quote}}\)


Well let me give you an alternative and very short method. I am going to continue from this step of @ikram002p :

\(\large{\sum_{j=1}^{5}(4j-16)=4\sum_{j=1}^{5}(j)-\sum_{j=1}^{5}(16)}\)

Answer 25

See:

observe the first summation:

\[\large{\sum_{j = 1}^{5}(j)}\]

It is like adding first 5 natural numbers. Now, I am providing you a formula:

sum of first n natural numbers = \(\cfrac{n(n+1)}{2}\)

Its very easy to remember.

Answer 26

So, sum of first 5 natural numbers = \(\cfrac{5(5+1)}{2}\)

Can you solve this and tell me the value ?

Answer 27

@sapphirelearner1995 ?

Answer 28

\[30/2=15\]

Answer 29

Good.

Now, replace this value by first summation.

Now, for the second summation:

\(\large{\sum_{j = 1}^{5} (16)}\)

Here, 16 is a constant and does not depend on j. So, I can take it outside and I will be left with:

\(\large16{\sum_{j = 1}^{5} (1)}\)

Now, this is equivalent to first adding 1, 5 times and then multiplying the answer by 16.

So, can you find this summation ?