In an AP, the sum of the first ten terms is -150 and the sum of its next ten terms is -550. Find AP.
Last updated date: 13th Mar 2023
•
Total views: 305.7k
•
Views today: 5.85k
Answer
305.7k+ views
Hint: Sum of n terms of AP ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$.
Given sum of first ten terms = $ - 150$
We know that Sum of n terms of AP ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$
Therefore,
$\
\Rightarrow {S_{10}} = \dfrac{{10}}{2}[2a + (10 - 1)d] \\
\Rightarrow - 150 = 5[2a + 9d] \\
\Rightarrow - 150 = 10a + 45d \to (1) \\
\ $
And also given that sum of next ten terms =$ - 550$ (which also includes sum of first ten terms value which has to be removed)
Sum of next ten terms = $ - 550$
$ \Rightarrow $$ - 150$$ - 550$ = $\dfrac{{20}}{2}[2a + (20 - 1)d]$
$\
\Rightarrow - 700 = 10(2a + 19d) \\
\Rightarrow - 70 = 2a + 19d \to (2) \\
\ $
Multiplying equation $(2) \times 5$ then we get
$ \Rightarrow - 350 = 10a + 95d \to (3)$
On solving $(1)\& (3)$ we get
$d = - 4$
Now by substituting $'d'$ value in equation $(1)$ we get
$\
\Rightarrow - 150 = 10a + 45d \\
\Rightarrow - 150 = 10a + 45( - 4) \\
\Rightarrow 10a = - 150 + 180 \\
\Rightarrow 10a = 30 \\
\Rightarrow a = 3 \\
\ $
Hence we got the value $a = 3,d = - 4$
We know that for an AP series $'a'$ be the first term and $'d'$ is the difference between the terms.
We also know that AP series will be of the form $a,a + d,a + 2d,a + 3d.......$
On substituting the $'a'$ and $'d'$ values
We get the values of series as 3,-1,-5,-9
Then the AP series will be $3, - 1, - 5, - 9....$
Note: In the above problem second condition i.e. sum of next ten terms includes sum of first 10 terms plus the other ten terms (where sum of first ten terms need to be subtracted from sum given for second condition) .Ignoring such simple condition will affect the answer.
Given sum of first ten terms = $ - 150$
We know that Sum of n terms of AP ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$
Therefore,
$\
\Rightarrow {S_{10}} = \dfrac{{10}}{2}[2a + (10 - 1)d] \\
\Rightarrow - 150 = 5[2a + 9d] \\
\Rightarrow - 150 = 10a + 45d \to (1) \\
\ $
And also given that sum of next ten terms =$ - 550$ (which also includes sum of first ten terms value which has to be removed)
Sum of next ten terms = $ - 550$
$ \Rightarrow $$ - 150$$ - 550$ = $\dfrac{{20}}{2}[2a + (20 - 1)d]$
$\
\Rightarrow - 700 = 10(2a + 19d) \\
\Rightarrow - 70 = 2a + 19d \to (2) \\
\ $
Multiplying equation $(2) \times 5$ then we get
$ \Rightarrow - 350 = 10a + 95d \to (3)$
On solving $(1)\& (3)$ we get
$d = - 4$
Now by substituting $'d'$ value in equation $(1)$ we get
$\
\Rightarrow - 150 = 10a + 45d \\
\Rightarrow - 150 = 10a + 45( - 4) \\
\Rightarrow 10a = - 150 + 180 \\
\Rightarrow 10a = 30 \\
\Rightarrow a = 3 \\
\ $
Hence we got the value $a = 3,d = - 4$
We know that for an AP series $'a'$ be the first term and $'d'$ is the difference between the terms.
We also know that AP series will be of the form $a,a + d,a + 2d,a + 3d.......$
On substituting the $'a'$ and $'d'$ values
We get the values of series as 3,-1,-5,-9
Then the AP series will be $3, - 1, - 5, - 9....$
Note: In the above problem second condition i.e. sum of next ten terms includes sum of first 10 terms plus the other ten terms (where sum of first ten terms need to be subtracted from sum given for second condition) .Ignoring such simple condition will affect the answer.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
