If the sum of first n terms of an AP is 1/7(3n^2+7n) then find its nth term and hence,write its 20th term.
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Ist term = s1 = 10/7
2nd term = s2-s1 = 16/7
3rd term = s3-s2 =22/7
Nth term = a+(n-1)d = 10/7 + (n-1)6/7
20th term = 124/7
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♠ ANSWER :
Sum of first n terms = 1 / 7 ( 3n^2 + 7 n )
Putting n = 1,
Sn = 1 / 7 ( 3 +7 ) = 10 / 7.
S1 = 10 / 7
Putting n = 2,
S2 = 1 / 7 ( 3 *2 ^2 + 7 *2 )
S2 = 1 / 7 ( 12 + 14 )
S2 = 1 / 7 ( 26 )
S2 = 26 / 7.
S2 - S1 = a2
26 / 7 - 10 /7 = 16 / 7.
S1 = a1 = 10 / 7.
AP = 10 / 7, 16 /7.
Common difference, d = a2 - a1 = 16 /7 - 10 /7
d = 6 /7.
Nth Term ,
An = a+ ( n - 1 ) d
An = 10 /7 + ( n - 1 ) 6 /7
An = 10 / 7 + 6n /7 - 6 / 7
An = 4 / 7 + 6n / 7.
⏺️An = ( 6n + 4) / 7
20th term,
A 20 = 10 / 7 + ( 20 - 1 ) 6 / 7
A 20 = 10 / 7 + 114 / 7
✔️A 20 = 124 / 7.
Sum of first n terms = 1 / 7 ( 3n^2 + 7 n )
Putting n = 1,
Sn = 1 / 7 ( 3 +7 ) = 10 / 7.
S1 = 10 / 7
Putting n = 2,
S2 = 1 / 7 ( 3 *2 ^2 + 7 *2 )
S2 = 1 / 7 ( 12 + 14 )
S2 = 1 / 7 ( 26 )
S2 = 26 / 7.
S2 - S1 = a2
26 / 7 - 10 /7 = 16 / 7.
S1 = a1 = 10 / 7.
AP = 10 / 7, 16 /7.
Common difference, d = a2 - a1 = 16 /7 - 10 /7
d = 6 /7.
Nth Term ,
An = a+ ( n - 1 ) d
An = 10 /7 + ( n - 1 ) 6 /7
An = 10 / 7 + 6n /7 - 6 / 7
An = 4 / 7 + 6n / 7.
⏺️An = ( 6n + 4) / 7
20th term,
A 20 = 10 / 7 + ( 20 - 1 ) 6 / 7
A 20 = 10 / 7 + 114 / 7
✔️A 20 = 124 / 7.
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