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

<p><b>Step 1: Breaking the sequence into two parts</b></p> <p>The given sequence is $1+3+5^2+7+9^2+\ldots$ up to 40 terms. Let's group the terms as follows:</p> <p>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 ($3, 7, 11, ...$).</p> <p><b>Step 2: Counting the terms in each group</b></p> <p>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.</p> <p><b>Step 3: Sums for each group</b></p> <p>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.</p> <p>The other numbers are $3, 7, 11, ..., (4k-1)$ for $k$ from 1 to 20.</p> <p><b>Step 4: Write these as formulas</b></p> <p>Total sum = Sum of squares part $+ $ Sum of ordinary numbers part.</p> <p>The sum of the squared numbers is $\sum_{k=1}^{20} (4k-3)^2$.</p> <p>The sum of the other numbers (arithmetic series): $3 + 7 + 11 + ...$ for 20 terms. </p> <p>The first term is 3, the last term is $3 + (20-1)\times4 = 3 + 76 = 79$. So, the sum is $\frac{20}{2} [3 + 79] = 10 \times 82 = 820$.</p> <p><b>Step 5: Expanding and simplifying</b></p> <p>Expand $(4k-3)^2$: $(4k-3)^2 = 16k^2 - 24k + 9$</p> <p>So, sum up these for $k=1$ to $20$: $16 \sum k^2 - 24 \sum k + 9 \times 20$</p> <p>We know formulas: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$<br> $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$</p> <p>Plug in $n=20$: $\sum_{k=1}^{20} k^2 = \frac{20 \times 21 \times 41}{6}$<br> $\sum_{k=1}^{20} k = \frac{20 \times 21}{2}$</p> <p>Now, sum up:</p> <p> $= 16 \left(\frac{20 \times 21 \times 41}{6}\right) - 24\left(\frac{20 \times 21}{2}\right) + 180 + 820$ </p> <p><b>Step 6: Calculating</b></p> <p> $16 \left(\frac{20 \times 21 \times 41}{6}\right) = 16 \times 2870 = 45920$<br> $24 \left(\frac{20 \times 21}{2}\right) = 24 \times 210 = 5040$<br> $9 \times 20 = 180$<br> Sum of the other numbers = $820$ </p> <p> So, total = $45920 - 5040 + 180 + 820$<br> $= 45920 - 5040 = 40880$<br> $40880 + 180 = 41060$<br> $41060 + 820 = 41880$ </p> <p><b>Final Answer</b></p> <p>The sum of the 40 terms is $41880$.</p>