For c|| n∈N∑i=0n(4i+3)=2n3+5n+3 prove

tripiverded9

tripiverded9

Answered

2021-12-30

For c|| nNi=0n(4i+3)=2n3+5n+3 prove

Answer & Explanation

Suhadolahbb

Suhadolahbb

Expert

2021-12-31Added 32 answers

Step 1
We can write summation as
i=0n(4i+3)=4(1+2+3++n)+3n+3
Step 2
we know sum of n terms of A.P. is nn+12
therefore =4n(n+1)2+3n+3
=2n2+5n+3 proved
Lynne Trussell

Lynne Trussell

Expert

2022-01-01Added 32 answers

Step 1
To prove, i=0n(4i+3)=2n2+5n+3
Step 2
Simplification:
Taking left hand side,
i=0n(4i+3)=i=0n4i+i=0n3
=4i=0ni+3i=0n1
=4(0+1+2+3++n)+3(1+1+1++1((n+1)×))
=4(n(n+1)2)+3(n+1) [Using 1+2+3++n=n(n+1)2]
=2n(n+1)+3(n+1)
=2n2+2n+3n+3
=2n2+5n+3
=Right hand side.
Hence proved.
user_27qwe

user_27qwe

Expert

2022-01-05Added 230 answers

i=0n(4i+3)=2n2+5n+3i=0n(4i+3)=i=0n4i+i=0n3=4i=0ni+3i=0n1=4(0+1+2+3+...+n)+3(1+1+1+...+1((n+1) times))=4(n(n+1)2)+3(n+1) [Using 1+2+3+...+n=n(n+1)2]=2n(n+1)+3(n+1)=2n2+2n+3n+3=2n2+5n+3
Answer: Right hand side.

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