The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167 . If
the sum of the first ten terms of this AP is 235 , find the sum of its first twenty terms.

Question

The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167 . If the sum of the first ten terms of this AP is 235 , find the sum of its first twenty terms.

UpStudy AI · Accepted Answer

Let the first term be $$ a $$ and the common difference be $$ d $$. The sum of the first $$ n $$ terms of an arithmetic progression (AP) is given by

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

**Step 1. Expressing the Given Sums**

For the first $$5$$ terms:

$$
S_5 = \frac{5}{2}\left[2a + 4d\right] = \frac{5}{2} \cdot 2(a + 2d) = 5(a + 2d).
$$

For the first $$7$$ terms:

$$
S_7 = \frac{7}{2}\left[2a + 6d\right] = \frac{7}{2} \cdot 2(a + 3d) = 7(a + 3d).
$$

According to the problem, the sum of these two sums is

$$
S_5 + S_7 = 5(a + 2d) + 7(a + 3d) = 167.
$$

Expanding this:

$$
5a + 10d + 7a + 21d = 12a + 31d = 167. \tag{1}
$$

For the first $$10$$ terms:

$$
S_{10} = \frac{10}{2}\left[2a + 9d\right] = 5(2a + 9d) = 10a + 45d.
$$

It is given that

$$
10a + 45d = 235. \tag{2}
$$

**Step 2. Solving the Equations**

We have the system of equations:

$$
\begin{cases}
12a + 31d = 167, \
10a + 45d = 235.
\end{cases}
$$

To eliminate $$a$$, multiply equation $$(1)$$ by $$5$$ and equation $$(2)$$ by $$6$$:

$$
\begin{aligned}
5(12a + 31d) &= 5 \times 167 \quad \Longrightarrow \quad 60a + 155d = 835, \
6(10a + 45d) &= 6 \times 235 \quad \Longrightarrow \quad 60a + 270d = 1410.
\end{aligned}
$$

Subtract the first new equation from the second:

$$
(60a + 270d) - (60a + 155d) = 1410 - 835,
$$

which simplifies to

$$
115d = 575.
$$

Thus,

$$
d = \frac{575}{115} = 5.
$$

Now substitute $$ d = 5 $$ into equation $$(2)$$:

$$
10a + 45(5) = 235 \quad \Longrightarrow \quad 10a + 225 = 235.
$$

Solving for $$ a $$:

$$
10a = 235 - 225 = 10 \quad \Longrightarrow \quad a = 1.
$$

**Step 3. Finding the Sum of the First Twenty Terms**

The sum $$ S_{20} $$ is

$$
S_{20} = \frac{20}{2}\left[2a + 19d\right] = 10\left[2(1) + 19(5)\right].
$$

Calculate inside the bracket:

$$
2 + 95 = 97.
$$

Thus,

$$
S_{20} = 10 \times 97 = 970.
$$

$$
\boxed{970}
$$
The sum of the first twenty terms is 970.

The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167 . If
the sum of the first ten terms of this AP is 235 , find the sum of its first twenty terms.

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