2x+1, 4x-1, 5x+1 are in an AS. Write the terms.
Answers
Answer:
GiveN
2x + 1, 4x -1, 5x +1 are in AP
✏ To FinD:
Find the value of x....?
Find the position of 195 in the sequence.
━━━━━━━━━━━━━━━━━━━━
✳ How to solve?
For solving the above question we need to know the relation between the terms by it's common difference.
Let three numbers a,b,c are in AP
Then, b - a = c - b = Common difference
➝ b - a = c - b
➝ 2b = a + c
✒ So, 2(second term) = First term + Third term
In this way, we can find x, After that we will get out first term and common difference. We have the nth term, So for finding which term it is, W can use....
\large{ \boxed{ \rm{t_n = a + (n - 1)d}}}
t
n
=a+(n−1)d
Where, tn is the nth term, a is first term, n is no. of terms.and d is the common difference.
✒ So, let's solve this question...
━━━━━━━━━━━━━━━━━━━━
✳ Solution:
Since, 2x + 1, 4x -1, 5x +1 are in AP
\begin{gathered}\large{ \rm{ \therefore{2(4x - 1) = (2x + 1) + (5x + 1)}}} \\ \\ \large{ \rm{ \longrightarrow \: 8x - 2 = 7x + 2}} \\ \\ \large{ \rm{ \longrightarrow \: 8x - 7x = 2 + 2}} \\ \\ \large{ \rm{ \longrightarrow \: \boxed{ \red{ \rm{ x = 4}}}}}\end{gathered}
∴2(4x−1)=(2x+1)+(5x+1)
⟶8x−2=7x+2
⟶8x−7x=2+2
⟶
x=4
Now,
First term = 2(4) + 1 = 9
Common difference = Second term- first term
= 4x - 1 - (2x +1)
= 4x - 1 - 2x - 1
= 2x - 2
Putting the value of x
= 2(4) - 2 = 6
nth term = 195
By using formula,
\begin{gathered}\large{ \rm{ \longrightarrow \: 195 = 9 + (n - 1)6}} \\ \\ \large{ \rm{ \longrightarrow \: 186 = 6(n - 1)}} \\ \\ \large{ \rm{ \longrightarrow \: n - 1 = \cancel{\frac{186}{6} }}} \\ \\ \large{ \rm{ \longrightarrow \: n - 1 = 31}} \\ \\ \large{ \rm{ \longrightarrow \: \boxed{ \red{ \rm{n = 32}}}}}\end{gathered}
⟶195=9+(n−1)6
⟶186=6(n−1)
⟶n−1=
6
186
⟶n−1=31
⟶
n=32
✏ Hence, 195 is the 32th term of the AP
\large{ \therefore{ \underline{ \underline{ \purple{ \rm{Hence \: solved \: \dag}}}}}}∴
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✳ Required Answer:
✏ GiveN
2x + 1, 4x -1, 5x +1 are in AP
✏ To FinD:
Find the value of x....?
Find the position of 195 in the sequence.
━━━━━━━━━━━━━━━━━━━━
✳ How to solve?
For solving the above question we need to know the relation between the terms by it's common difference.
Let three numbers a,b,c are in AP
Then, b - a = c - b = Common difference
➝ b - a = c - b
➝ 2b = a + c
✒ So, 2(second term) = First term + Third term
In this way, we can find x, After that we will get out first term and common difference. We have the nth term, So for finding which term it is, W can use....
Where, tn is the nth term, a is first term, n is no. of terms.and d is the common difference.
✒ So, let's solve this question...
━━━━━━━━━━━━━━━━━━━━
✳ Solution:
Since, 2x + 1, 4x -1, 5x +1 are in AP
Now,
First term = 2(4) + 1 = 9
Common difference = Second term- first term
= 4x - 1 - (2x +1)
= 4x - 1 - 2x - 1
= 2x - 2
Putting the value of x
= 2(4) - 2 = 6
nth term = 195
By using formula,
✏ Hence, 195 is the 32th term of the AP