Solve this: Q 20 Find the nth term and the sum of n terms of the series 13 7+17 - Maths - Sequences and Series

Question

Solve this:
Q 20 Find the n t h term and the sum of n terms of the series 1 3 7
+ 1 7 11 + 1 11 15 +

Aarushi Mishra · Accepted Answer

an=13 7 , 17 11 , 111 15 ,
Note:bn=3 ,7 ,11 , 15, 
 forms an A P
 with first term b=3 and commmon difference d=4bn=b+n-1d=3+n-1×4bn=4n-1an=13
7 , 17 11 , 111 15 ,  =1b1
b2 , 1b2 b3 , 1b3 b4 ,
Therefore we havea1=1b1 b2a2=1b2 b3a3=1b3
b4Following the patternan=1bn bn+1Sn=13 7 +17
11 +111 15 +
+an=∑r=1n ar∑r=1n ar=∑r=1n 1br
br+1=∑r=1n1br+1-br×br+1-brbr
br+1br+1-br=4 as bn is an A P
 of common difference 4=∑r=1n 14br+1-brbr
br+1=∑r=1n 14br+1br br+1-brbr
br+1=∑r=1n 141br-1br+1=14∑r=1n 1br-1br+1=141b1-1b2+1b2-1b3+1b3-1b4+
+1bn-1bn+1=141b1-1b2+1b2-1b3+1b3-1b4+
+1bn-1bn+1=141b1-1bn+1=1413-14n+1-1=1413-14n+3 =144n+3-334n+3 =144n34n+3 =n34n+3

Solve this: Q.20. Find the n t h term and the sum of n terms of the series 1 3 . 7 + 1 7 . 11 + 1 11 . 15 + . . . . . . . .

Solve this:
Q.20. Find the $n^{t h}$ term and the sum of n terms of the series $\frac{1}{3.7} + \frac{1}{7.11} + \frac{1}{11.15} + . . . . . . . .$