Let Sn denote the sum of the first n terms of an arithmetic progression. If S10 = 390 and the ratio

Q: Wait, why can't we just use $S_{15} - S_5 = S_{10}$? Isn't that the same thing?

That is a very common trap! $S_{10}$ is the sum of the first 10 terms ($T_1$ to $T_{10}$). However, $S_{15} - S_5$ is the sum of terms from $T_6$ to $T_{15}$. These are two completely different sets of terms. Always visualize the range of terms you are summing!

Q: Is there a faster way to calculate $S_{15} - S_5$ without finding $a$ and $d$?

While you could potentially manipulate the formulas to find the sum directly, finding $a$ and $d$ is the most robust and error-proof method. In JEE Advanced, clarity and accuracy are more important than finding a 'shortcut' that might lead to a sign error.

Q: Why did we cross-multiply the ratio instead of solving for $a$ in terms of $d$ immediately?

Cross-multiplying is the standard, most reliable way to handle ratios in equations. It transforms a fractional relationship into a simple linear equation, which is much easier to work with when you have a system of equations. It minimizes the chance of making algebraic mistakes.

Question

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$, then $S_{15} - S_5$ is equal to:

Options:

(A) 800

(B) 890

(C) 790

(D) 690

Sigma Percentile · Accepted Answer

### The Architecture of Sequences: A Journey into Arithmetic Progressions

Imagine you are standing before a grand staircase. Each step you take is higher than the last by a constant amount; this is the essence of an Arithmetic Progression (AP).

In this problem, we are uncovering the hidden structure of this staircase. We are given two clues: the sum of the first ten steps, $S_{10} = 390$, and the ratio of the tenth step to the fifth step, which is $15:7$.

Our mission is to find the sum of the steps from the sixth to the fifteenth, which is mathematically represented as $S_{15} - S_5$.

### The Blueprint: Defining Our Variables

Every journey begins with a definition. Let the first step be $a$ and the constant height added at each step be $d$.

Our toolkit consists of two fundamental formulas:

$$S_n = \frac{n}{2}[2a + (n-1)d]$$

$$T_n = a + (n-1)d$$

These are our compass and map for navigating the sequence.

### The Detective Work: Setting the Constraints

We are told that $S_{10} = 390$. By substituting $n=10$ into our sum formula:

$$S_{10} = \frac{10}{2}[2a + (10-1)d] = 390$$

Simplifying this, we get $5[2a + 9d] = 390$, which reduces to the linear equation:

$$2a + 9d = 78 \quad 	ext{(Equation 1)}$$

Next, we look at the ratio of the tenth term to the fifth term: $\frac{T_{10}}{T_5} = \frac{15}{7}$. Using our general term formula:

$$\frac{a + 9d}{a + 4d} = \frac{15}{7}$$

By cross-multiplying, we get $7(a + 9d) = 15(a + 4d)$, which expands to $7a + 63d = 15a + 60d$. Rearranging the terms, we find the relationship:

$$3d = 8a \quad 	ext{(Equation 2)}$$

### The Intersection: Solving the System

Now, we have a system of two linear equations. We can substitute Equation 2 into Equation 1.

Since $3d = 8a$, then $9d = 3 	imes (3d) = 3 	imes (8a) = 24a$. Substituting this into $2a + 9d = 78$:

$$2a + 24a = 78 \Rightarrow 26a = 78$$

Thus, $a = 3$. With $a$ known, we find $d$ using $3d = 8(3) = 24$, so $d = 8$. Our staircase starts at $3$ and grows by $8$ with every step.

### The Final Calculation: Reaching the Summit

The question asks for $S_{15} - S_5$. Visually, $S_{15}$ is the sum of the first 15 terms, and $S_5$ is the sum of the first 5; subtracting them leaves us with the sum of terms from $T_6$ to $T_{15}$.

First, we calculate $S_{15}$:

$$S_{15} = \frac{15}{2}[2(3) + (15-1)8] = \frac{15}{2}[6 + 112] = \frac{15}{2}[118] = 15 	imes 59 = 885$$

Next, we calculate $S_5$:

$$S_5 = \frac{5}{2}[2(3) + (5-1)8] = \frac{5}{2}[6 + 32] = \frac{5}{2}[38] = 5 	imes 19 = 95$$

Finally, we perform the subtraction:

$$S_{15} - S_5 = 885 - 95 = 790$$

We have successfully navigated the staircase and reached the summit. The final answer is **790**.

Animated Solution for Mathematics - Sequence and Series: Let $S_{n}$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$ , then $S_{15} - S_{5}$ is equal to:

Select Answer:

Visualized Solution

Visualizing the Arithmetic Progression

Setting up $S_{10}$

Simplifying the First Equation

Setting up the Ratio $T_{10} : T_{5}$

Cross-Multiplying the Ratio

Finding the Relation between $a$ and $d$

Solving for the First Term $a$

Solving for the Common Difference $d$

Visualizing the Target: $S_{15} - S_{5}$

Calculating $S_{15}$

Calculating $S_{5}$

The Final Answer

The Sigma Insight: Arithmetic Progression (A.P.)

The Architecture of Sequences

The Blueprint

The Detective Work

The Intersection

The Final Calculation

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Select Answer:

Visualized Solution

Visualizing the Arithmetic Progression

Setting up .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S10​

Simplifying the First Equation

Setting up the Ratio .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T10​:T5​

Cross-Multiplying the Ratio

Finding the Relation between .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Solving for the First Term .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Solving for the Common Difference .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Visualizing the Target: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S15​−S5​

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S15​

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S5​