Find out the answer to "For c|| n∈N∑i=0n(4i+3)=2n3+5n+3 prove"

High School
Algebra
Algebra I
Algebra foundations

tripiverded9

Answered question

2021-12-30

For c||

n \in N \sum_{i = 0}^{n} (4 i + 3) = 2 n^{3} + 5 n + 3

prove

Answer & Explanation

Suhadolahbb

Beginner2021-12-31Added 32 answers

Step 1
We can write summation as

\sum_{i = 0}^{n} (4 i + 3) = 4 (1 + 2 + 3 + \dots + n) + 3 n + 3

Step 2
we know sum of n terms of A.P. is

n \frac{n + 1}{2}

therefore

= 4 \frac{n (n + 1)}{2} + 3 n + 3

= 2 n^{2} + 5 n + 3

proved

Lynne Trussell

Beginner2022-01-01Added 32 answers

Step 1
To prove,

\sum_{i = 0}^{n} (4 i + 3) = 2 n^{2} + 5 n + 3

Step 2
Simplification:
Taking left hand side,

\sum_{i = 0}^{n} (4 i + 3) = \sum_{i = 0}^{n} 4 i + \sum_{i = 0}^{n} 3

= 4 \sum_{i = 0}^{n} i + 3 \sum_{i = 0}^{n} 1

= 4 (0 + 1 + 2 + 3 + \dots + n) + 3 (1 + 1 + 1 + \dots + 1 ((n + 1) \times))

= 4 (\frac{n (n + 1)}{2}) + 3 (n + 1) [U \sin g 1 + 2 + 3 + \dots + n = \frac{n (n + 1)}{2}]

= 2 n (n + 1) + 3 (n + 1)

= 2 n^{2} + 2 n + 3 n + 3

= 2 n^{2} + 5 n + 3

= Right hand side.

Hence proved.

user_27qwe

Skilled2022-01-05Added 375 answers

$\begin{matrix} \sum_{i = 0}^{n} (4 i + 3) = 2 n^{2} + 5 n + 3 \\ \sum_{i = 0}^{n} (4 i + 3) = \sum_{i = 0}^{n} 4 i + \sum_{i = 0}^{n} 3 \\ = 4 \sum_{i = 0}^{n} i + 3 \sum_{i = 0}^{n} 1 \\ = 4 (0 + 1 + 2 + 3 + . . . + n) + 3 (1 + 1 + 1 + . . . + 1 ((n + 1) t i m e s)) \\ = 4 (\frac{n (n + 1)}{2}) + 3 (n + 1) [U s i n g 1 + 2 + 3 + . . . + n = \frac{n (n + 1)}{2}] \\ = 2 n (n + 1) + 3 (n + 1) \\ = 2 n^{2} + 2 n + 3 n + 3 \\ = 2 n^{2} + 5 n + 3 \end{matrix}$
Answer: Right hand side.

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