### The sum of the first six terms of an AP is 9 and then fourth...

The sum of the first six terms of an AP is 9 and then fourth term is 2. Find the seventh term of the AP?

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Hykay
7 months ago
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• Hykay: the solutions are in the image above
7 months ago
• Hykay: there are some mistakes in the first solution, so I've corrected it here
7 months ago
ROBERT
7 months ago
This question will definitely lead to a simultaneous equation. Kindly observe:

The sum of an nth term= Sn= ½(n)(2a + (n - 1)d), where a= first term and d= common difference in an Arithmetic Progression(AP).

Now, the sum of the first six terms= S6= ½(6)(2a + (6- 1)d)= 9
= 3(2a + 5d)= 9
= 3(2a) + 3(5d)= 9
= 6a + 15d= 9
Divide all sides by three and we will have:
2a + 5d= 3 •••> (1)

Another, the fourth term is 2.
Recall, Tn= a + (n - 1)d
So, T4= a + (4 - 1)d= 2
= a + 3d= 2 •••>(2)

2a + 5d= 3 •••>(1)
a + 3d= 2 •••>(2)
-------------------------- Using the elimination method,
2a + 5d= 3 •••>(1) × 1
a + 3d= 2 •••>(2) × - 2
---------------------------
2a + 5d= 3 •••>(3)
-2a - 6d= - 4 •••>(4)
-------------------- Add up equation (3) and (4)
- d= - 1
d= 1

From equation (2),
a + 3d= 2 (d= 1)
a + 3(1)= 2
a + 3= 2
a= 2 - 3
a= -1

Thus, a= -1; d= 1

The seventh term of the AP= T7= a + 6d
= (-1) + 6(1)
= -1 + 6
= 5

Therefore, the value of the seventh term is 5.
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