Suppose a1, a2, 2, a3, a4 be in an arithmetico-geometric progression. If the common ratio of the cor

Q: Wait, why is the third term $4(A+2d)$? Shouldn't it be $2(A+2d)$?

Great question! Remember the general term formula is $T_n = (A + (n-1)d) \cdot r^{n-1}$. For the third term, $n=3$, so the geometric part is $r^{3-1} = r^2$. Since $r=2$, $r^2 = 4$. That is why the third term is $4(A+2d)$.

Q: Can I use the standard formula for the sum of an AGP?

You could, but for a small number of terms like $n=5$, it is often faster and less prone to error to simply write out the five terms and sum them manually. It helps you avoid misapplying the formula's conditions.

Q: I got $A=0$. Is that allowed? Did I make a mistake?

Absolutely not a mistake! In sequences, the first term can certainly be zero. It just means your arithmetic progression starts at zero. Trust your algebra—if the steps are correct, the result is correct.

Question

Suppose $a_1, a_2, 2, a_3, a_4$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $\frac{49}{2}$, then $a_4$ is equal to ______________

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the intersection of two distinct mathematical paths. On one side, we have the steady, additive rhythm of an Arithmetic Progression (AP). On the other, the explosive, multiplicative growth of a Geometric Progression (GP).

When these two worlds collide, they form an Arithmetico-Geometric Progression (AGP). We are tasked with finding the hidden values of a sequence: $a_1, a_2, 2, a_3, a_4$.

We are given that the common ratio of the geometric part is $r=2$, and the sum of these five terms is $\frac{49}{2}$. Our mission is to find $a_4$.

### The Master Key

To unlock this sequence, we use the general term formula for an AGP:
$$T_n = (A + (n-1)d) \cdot r^{n-1}$$

Here, $A$ is the first term of the AP, $d$ is the common difference, and $r$ is the common ratio. This formula acts as the bridge between the two worlds.

Let us map out our five terms using this key:
- For $n=1$: $T_1 = A \cdot 2^0 = A$
- For $n=2$: $T_2 = (A+d) \cdot 2^1 = 2(A+d)$
- For $n=3$: $T_3 = (A+2d) \cdot 2^2 = 4(A+2d)$
- For $n=4$: $T_4 = (A+3d) \cdot 2^3 = 8(A+3d)$
- For $n=5$: $T_5 = (A+4d) \cdot 2^4 = 16(A+4d)$

### The Anchor Point

We are given a vital piece of information: the third term is $2$. Using our mapping, we have:
$$4(A+2d) = 2$$

Dividing both sides by $4$, we get $A+2d = \frac{1}{2}$. This is our anchor, providing a direct relationship between $A$ and $d$:
$$A = \frac{1}{2} - 2d$$

### The Grand Summation

Now, we tackle the sum of all five terms:
$$S_5 = A + 2(A+d) + 4(A+2d) + 8(A+3d) + 16(A+4d) = \frac{49}{2}$$

Grouping the terms by their variables, we collect all $A$ terms:
$$A(1 + 2 + 4 + 8 + 16) = 31A$$

Collecting all $d$ terms:
$$d(0 + 2 + 8 + 24 + 64) = 98d$$

Our equation becomes $31A + 98d = \frac{49}{2}$. Substituting our anchor $A = \frac{1}{2} - 2d$ into this equation:
$$31\left(\frac{1}{2} - 2d\right) + 98d = \frac{49}{2}$$

Expanding this, we get:
$$\frac{31}{2} - 62d + 98d = \frac{49}{2}$$

Simplifying the $d$ terms:
$$36d = \frac{49}{2} - \frac{31}{2} = \frac{18}{2} = 9$$

Thus, $36d = 9$, which yields $d = \frac{9}{36} = \frac{1}{4}$.

### The Resolution

With $d = \frac{1}{4}$ in hand, we find $A$:
$$A = \frac{1}{2} - 2\left(\frac{1}{4}\right) = \frac{1}{2} - \frac{1}{2} = 0$$

The first term of our AP is zero. Finally, we calculate our target, $a_4$, which corresponds to the fifth term of the sequence ($T_5$):
$$a_4 = 16(A + 4d) = 16\left(0 + 4 \cdot \frac{1}{4}\right) = 16(1) = 16$$

The final answer is **16**.

Enter Numerical Value:

Visualized Solution

Given AGP Sequence

General Term of AGP

Expressing the Terms

Relation from the Third Term

Sum of the 5 Terms

Expanding the Summation

Solving for $d$

Calculating the Common Difference

Calculating the First Term

Calculating $a_{4}$

Final Answer

The Sigma Insight: Arithmetic-Geometric Progression (A.G.P.)

Analyzing the Setup

The Master Key

The Anchor Point

The Grand Summation

The Resolution

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