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

Q: Why can't we just find one general term for the whole series?

The series is piecewise and alternating. It switches between a squared term and a linear term. Trying to force a single formula would lead to a complex, non-standard expression. Splitting it into two sub-series is the most efficient and mathematically sound approach.

Q: How do I know how many terms go into each sub-series?

Since the total number of terms is $40$ and the pattern alternates perfectly (one squared, one linear), we simply divide the total count by $2$. Thus, each sub-series will contain exactly $20$ terms.

Q: Is there a shortcut to avoid expanding the quadratic term?

While you could potentially use properties of summations, expanding $(4r-3)^2$ is the most reliable method. It transforms the expression into a standard polynomial form, allowing you to directly apply the well-known summation formulas for $\sum r^2$, $\sum r$, and constants.

Question

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

Options:

(A) 43890

(B) 41880

(C) 33980

(D) 40870

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the mathematical realm. Today, we are going to dissect a series that might look intimidating at first glance: $1 + 3 + 5^2 + 7 + 9^2 + \dots$ up to $40$ terms.

When you see a series like this, your first instinct might be to panic, but I want you to take a deep breath. Mathematics is not about brute force; it is about pattern recognition.

Look closely at the rhythm: $1$, then $3$, then $5^2$, then $7$, then $9^2$. It is a dance between squared numbers and linear numbers. If we try to find one general term for this entire sequence, we will end up in a labyrinth. Instead, we use the strategy of **Divide and Conquer**.

### The Divide and Conquer: Splitting the Series

Since we have $40$ terms in total and the pattern alternates perfectly, we can split this into two distinct sub-series. Each sub-series will have exactly $20$ terms.

Our first sub-series, which we will call $S_1$, consists of the odd-positioned terms: $1^2 + 5^2 + 9^2 + \dots$ (up to $20$ terms). Note that $1$ is simply $1^2$.

Our second sub-series, $S_2$, consists of the even-positioned terms: $3 + 7 + 11 + \dots$ (up to $20$ terms). By separating them, we have turned one impossible problem into two very manageable ones.

### The General Term: Finding the Soul of the Series

Let us look at the bases of the first sub-series: $1, 5, 9, \dots$. This is a classic Arithmetic Progression (AP) where the first term $a = 1$ and the common difference $d = 4$.

The general term for these bases is $a + (r-1)d = 1 + (r-1)4 = 4r - 3$. Since our terms are squared, the general term for $S_1$ is $T_{1,r} = (4r-3)^2$.

Now, for the second sub-series: $3, 7, 11, \dots$. This is another AP with $a = 3$ and $d = 4$. Its general term is $T_{2,r} = 3 + (r-1)4 = 4r - 1$. We have successfully captured the essence of both series.

### The Algebraic Symphony: Summation and Expansion

Now, we combine them into a single summation:

$$S = \sum_{r=1}^{20} [(4r-3)^2 + (4r-1)]$$

Before we can calculate this, we must expand the quadratic term. Using the identity $(a-b)^2 = a^2 - 2ab + b^2$, we expand $(4r-3)^2$ to get $16r^2 - 24r + 9$.

Adding the linear term $(4r-1)$, our expression becomes $16r^2 - 24r + 9 + 4r - 1$, which simplifies beautifully to $16r^2 - 20r + 8$. We are now looking at the sum:

$$S = \sum_{r=1}^{20} (16r^2 - 20r + 8)$$

### The Final Calculation: Reaching the Summit

We can now distribute the summation:

$$S = 16 \sum_{r=1}^{20} r^2 - 20 \sum_{r=1}^{20} r + \sum_{r=1}^{20} 8$$

Using the standard formulas $\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$ with $n=20$, we calculate:

$$S = 16 \left( \frac{20 \cdot 21 \cdot 41}{6} \right) - 20 \left( \frac{20 \cdot 21}{2} \right) + 8(20)$$

This simplifies to $16(2870) - 20(210) + 160$, which results in $45920 - 4200 + 160$.

The final answer is **$41880$**. The beauty of this problem lies not just in the final number, but in the elegance of the process.

Animated Solution for Mathematics - Sequence and Series: $1 + 3 + 5^{2} + 7 + 9^{2} + \dots$ upto 40 terms is equal to

Select Answer:

Visualized Solution

Analyze the Series Pattern

Split the Series into Two Sub-series

General Term for the Squared Series

General Term for the Linear Series

Combine into a Single Summation

Expand the Quadratic Term

Simplify the Combined Expression

Apply Summation Formulas

Execute Final Calculations

Conclusion and Key Takeaway

The Sigma Insight: Sum of Special Series

Analyzing the Setup

The Divide and Conquer

The General Term

The Algebraic Symphony

The Final Calculation

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Animated Solution for Mathematics - Sequence and Series: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1+3+52+7+92+… upto 40 terms is equal to