1+3+5^2+7+9^2+ upto 40 terms is equal to

Question

$1+3+5^2+7+9^2+\ldots$ upto 40 terms is equal to

Accepted Answer

Breaking the sequence into two parts

The given sequence is $1+3+5^2+7+9^2+\ldots$ up to 40 terms. Let's group the terms as follows:

The terms are arranged like this: $1$, $3$, $5^2$, $7$, $9^2$, $11$, $13^2$, $15$, $17^2$, ... You can see that every third term is being squared, while others are just numbers. So, split the sequence into: all squared terms ($1^2, 5^2, 9^2, ...$) and all other numbers ($3, 7, 11, ...$).

Counting the terms in each group

Out of 40 terms, half will be squared terms and half will be non-squared terms. So we have 20 squared terms and 20 ordinary numbers.

Sums for each group

The squared numbers follow this pattern: $1^2, 5^2, 9^2, ...$, which can be written as $(4k-3)^2$ for $k$ from 1 to 20.

The other numbers are $3, 7, 11, ..., (4k-1)$ for $k$ from 1 to 20.

Write these as formulas

Total sum = Sum of squares part $+ $ Sum of ordinary numbers part.

The sum of the squared numbers is $\sum_{k=1}^{20} (4k-3)^2$.

The sum of the other numbers (arithmetic series): $3 + 7 + 11 + ...$ for 20 terms.

The first term is 3, the last term is $3 + (20-1)	imes4 = 3 + 76 = 79$. So, the sum is $\frac{20}{2} [3 + 79] = 10 	imes 82 = 820$.

Expanding and simplifying

Expand $(4k-3)^2$:

$(4k-3)^2 = 16k^2 - 24k + 9$

So, sum up these for $k=1$ to $20$:

$16 \sum k^2 - 24 \sum k + 9 	imes 20$

We know formulas:

$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

Plug in $n=20$:

$\sum_{k=1}^{20} k^2 = \frac{20 	imes 21 	imes 41}{6}$

$\sum_{k=1}^{20} k = \frac{20 	imes 21}{2}$

Now, sum up:

$= 16 \left(\frac{20 	imes 21 	imes 41}{6}ight) - 24\left(\frac{20 	imes 21}{2}ight) + 180 + 820$

Calculating

$16 \left(\frac{20 	imes 21 	imes 41}{6}ight) = 16 	imes 2870 = 45920$

$24 \left(\frac{20 	imes 21}{2}ight) = 24 	imes 210 = 5040$

$9 	imes 20 = 180$

Sum of the other numbers = $820$

So, total = $45920 - 5040 + 180 + 820$

$= 45920 - 5040 = 40880$

$40880 + 180 = 41060$

$41060 + 820 = 41880$

Final Answer

The sum of the 40 terms is $41880$.

Solutions · 1