How many terms of the A.P.
Answers
Given AP series is
It means,
↝ Let number of terms be n.
According to statement,
↝ Sum of n terms of AP series, = 300
We know that,
↝ Sum of n terms of an arithmetic sequence is,
Wʜᴇʀᴇ,
- Sₙ is the sum of n terms of AP.
- a is the first term of the sequence.
- n is the no. of terms.
- d is the common difference.
↝ On substituting the values, we get
Justification of double answer,
Note :-
Here, double answer indicates that sum of 26th terms to 36th terms are 0.
Number of terms from 26th term to 36th term = 11
Hence,
Hence, justify
Additional Information :-
↝ nᵗʰ term of an arithmetic sequence is,
Wʜᴇʀᴇ,
- aₙ is the nᵗʰ term.
- a is the first term of the sequence.
- n is the no. of terms.
- d is the common difference.
Answer:
\large\underline{\sf{Solution-}}
Solution−
Given AP series is
\rm :\longmapsto\:20, \: 19\dfrac{1}{3}, \: 18 \dfrac{2}{3}, \: - - - :⟼20,19
3
1
,18
3
2
,−−−
It means,
\rm :\longmapsto\:First \: term \: of \: an \: AP \: series, a = 20:⟼FirsttermofanAPseries,a=20
\rm :\longmapsto\:Common \: Difference, d = 19 \dfrac{1}{3} - 20:⟼CommonDifference,d=19
3
1
−20
\rm \: \: = \: \dfrac{58}{3} - 20=
3
58
−20
\rm \: \: = \: \dfrac{58 - 20}{3} =
3
58−20
\rm \: \: = \: \dfrac{ - 2}{3} =
3
−2
\rm :\longmapsto\:Common \: Difference, \: d = - \: \dfrac{2}{3} :⟼CommonDifference,d=−
3
2
↝ Let number of terms be n.
According to statement,
↝ Sum of n terms of AP series, = 300
We know that,
↝ Sum of n terms of an arithmetic sequence is,
\begin{gathered}\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}\end{gathered}
★
S
n
=
2
n
(2a+(n−1)d)
Wʜᴇʀᴇ,
Sₙ is the sum of n terms of AP.
a is the first term of the sequence.
n is the no. of terms.
d is the common difference.
↝ On substituting the values, we get
\rm :\longmapsto\:300 = \dfrac{n}{2}\bigg(2 \times 20 - \dfrac{2}{3} (n - 1) \bigg) :⟼300=
2
n
(2×20−
3
2
(n−1))
\rm :\longmapsto\:300 = \dfrac{n}{2}\bigg(40 - \dfrac{2}{3} (n - 1) \bigg) :⟼300=
2
n
(40−
3
2
(n−1))
\rm :\longmapsto\:300 = \dfrac{n}{2}\bigg(\dfrac{120 - 2n + 2}{3} \bigg) :⟼300=
2
n
(
3
120−2n+2
)
\rm :\longmapsto\:300 = \dfrac{n}{2}\bigg(\dfrac{122 - 2n }{3} \bigg) :⟼300=
2
n
(
3
122−2n
)
\rm :\longmapsto\:300 =n\bigg(\dfrac{61 - n }{3} \bigg) :⟼300=n(
3
61−n
)
\rm :\longmapsto\:900 = 61n - {n}^{2} :⟼900=61n−n
2
\rm :\longmapsto\: {n}^{2} - 61n + 900 = 0:⟼n
2
−61n+900=0
\rm :\longmapsto\: {n}^{2} - 36n - 25n + 900 = 0:⟼n
2
−36n−25n+900=0
\rm :\longmapsto\:n(n - 36) - 25(n - 36) = 0:⟼n(n−36)−25(n−36)=0
\rm :\longmapsto\:(n - 25)(n - 36) = 0:⟼(n−25)(n−36)=0
\bf\implies \:n = 25 \: \: \: or \: \: \: n = 36⟹n=25orn=36
Justification of double answer,
Note :-
Here, double answer indicates that sum of 26th terms to 36th terms are 0.
Number of terms from 26th term to 36th term = 11
\rm :\longmapsto\:d = - \: \dfrac{2}{3} :⟼d=−
3
2
\rm :\longmapsto\:a_{26} = a + 25d = 20 - \dfrac{50}{3} = \dfrac{10}{3} :⟼a
26
=a+25d=20−
3
50
=
3
10
Hence,
\rm :\longmapsto\:S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg):⟼S
n
=
2
n
(2a+(n−1)d)
\rm :\longmapsto\:S_{11}\:=\dfrac{11}{2} \bigg(2 \: \times \dfrac{10}{3} \: - \dfrac{2}{3} \:(11\:-\:1) \bigg):⟼S
11
=
2
11
(2×
3
10
−
3
2
(11−1))
\rm :\longmapsto\:S_{11}\:=\dfrac{11}{2} \bigg( \dfrac{20}{3} \: - \dfrac{2}{3} \:(10) \bigg):⟼S
11
=
2
11
(
3
20
−
3
2
(10))
\rm :\longmapsto\:S_{11}\:=\dfrac{11}{2} \bigg( \dfrac{20}{3} \: - \dfrac{20}{3} \bigg):⟼S
11
=
2
11
(
3
20
−
3
20
)
\rm :\longmapsto\:S_{11}\:=0:⟼S
11
=0
Hence, justify
Additional Information :-
↝ nᵗʰ term of an arithmetic sequence is,
\begin{gathered}\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}\end{gathered}
★
a
n
=a+(n−1)d
Wʜᴇʀᴇ,
aₙ is the nᵗʰ term.
a is the first term of the sequence.
n is the no. of terms.
d is the common difference.
Step-by-step explanation:
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