The sequence is
1,3,3,3,5,5,5,5,5,7,7,7,7,7,7,7,………………………….
To find nth term
First term = 1
Second term=3
Third term =3
Fourth term=3
Fifth term=5
(1^2)th term = 1
2^2 th term= 3
3^2 th term=5
4^2 th term=7
5^2 th term=9
………………
……………..
n^2 th term=(2n-1)
EXAMPLE 1
what is 125 th term ?
121th term=11^2 th term =((2 x 11)-1)=21
144th term=12^2 th term=(2x12)-1)=23
Hence from 122 onwards and to 144th term =23
Therefore 125th term is 23
In this way we can find any term of this sequence
EXAMPLE 2
What is 1001th term ?
1001 lies between 31^2 th term and 32^2 th term
31^2 th term=((2x31)-1)=61
There fore 962th onwards to 1024th term=63
Therefore 1001th term is 63
To find the sum of n terms
Sum of 1st term=1
Sum of first 2 terms=1+3=4
Sum of first 3 terms=1+3+3=7
Sum of first 4 terms=1+3+3+3=(1^2)+(3^2)=10
Sum of first 5 terms=(1^2)+(3^2)+5=15
Sum of first 6 terms=(1^2)+(3^2)+5+5=20
Sum of first 7 terms=(1^2)+(3^2)+5+5+5=25
Sum of first 8 terms=(1^2)+(3^2)+5+5+5+5=30
Sum of first 9 terms=(1^2)+(3^2)+(5^2)=35
Sum of first 10 terms=(1^2)+(3^2)+(5^2)+7=42
Sum of first 11 terms=(1^2)+(3^2)+(5^2)+7+7=49
Sum of first 12 terms=(1^2)+(3^2)+(5^2)+7+7+7=56
Sum of first 13 terms=(1^2)+(3^2)+(5^2)+7+7+7+7=63
Sum of first 14 terms=(1^2)+(3^2)+(5^2)+7+7+7+7+7=70
Sum of first 15 terms=(1^2)+(3^2)+(5^2)+7+7+7+7+7+7=77
Sum of first 16 terms=(1^2)+(3^2)+(5^2)+(7^2)=84
Sum of (1^2) terms=1
Sum of (2^2) terms=1^2+ 3^2
Sum of (3^2) terms=1^2+3^2+5^2
Sum of (4^2) terms=1^2+3^2+5^2+7^2
……………………………………………………………………
Sum of (n^2) terms=1^2+3^2+5^2+…………..+(2n-1)^2
= (n/3) x (4(n^2) - 1)
Using this formula we can find Sum of any terms of the sequence in the following way
EXAMPLE 1
What is the sum of first 125 terms
Sum of first 121 terms(11^2) = (11/3)((4x121)-1)=1771
122nd term=23 [can be calculated by the above first method]
Sum of first 125 terms = (1771)+(23+23+23+23)=(1771)+((125-121)x23)=1863
EXAMPLE 2
What is the sum of first 1001 terms
Sum of first 961 terms(31^2) = (31/3)((4x961)-1)=39711
962th term=63 [can be calculated by the above first method]
Sum of first 1001 terms = (39711)+((1001-961)x63)=42231
In this calculation may be some figure errors, but the method is correct
