Solution:

Given information:
- Mean of 13 observations = 14
- Mean of the first 7 observations = 12
- Mean of the last 7 observations = 16

To find: The 7th observation

Mean is calculated by summing all the observations and dividing by the number of observations. We can use this information to set up equations and solve for the unknowns.

Let's denote the sum of the first 7 observations as S1 and the sum of the last 7 observations as S2.

Equations:

1) Mean of 13 observations = (S1 + S2) / 13 = 14
2) Mean of the first 7 observations = S1 / 7 = 12
3) Mean of the last 7 observations = S2 / 7 = 16

From equation 2, we can solve for S1:
S1 = 7 * 12 = 84

From equation 3, we can solve for S2:
S2 = 7 * 16 = 112

Now we can substitute the values of S1 and S2 into equation 1 to solve for the sum of all 13 observations:

(S1 + S2) / 13 = 14
(84 + 112) / 13 = 14
196 / 13 = 14
14 = 14

This confirms that our values for S1 and S2 are correct.

To find the 7th observation, we need to find the sum of the first 6 observations and the sum of the last 6 observations, then subtract these sums from the sum of all 13 observations.

Let's denote the sum of the first 6 observations as S1_6 and the sum of the last 6 observations as S2_6.

Equations:

1) S1_6 / 6 = 84 - X (X represents the 7th observation, which is unknown)
2) S2_6 / 6 = 112 - X

To find the sum of the first 6 observations (S1_6), we can use the sum of the first 7 observations (S1) and subtract the 7th observation (X):

S1_6 = S1 - X
S1_6 = 84 - X

To find the sum of the last 6 observations (S2_6), we can use the sum of the last 7 observations (S2) and subtract the 7th observation (X):

S2_6 = S2 - X
S2_6 = 112 - X

Now we can substitute the values of S1_6 and S2_6 into equation 1 and equation 2 to solve for X:

S1_6 / 6 = 84 - X
S2_6 / 6 = 112 - X

(84 - X) / 6 = 84 - X
112 - X / 6 = 112 - X

Simplifying the equations:

84 - X = 6(84 - X)
112 - X = 6(112 - X)

Expanding the equations:

84 - X = 504 - 6X
112 - X = 672 - 6X

Rearranging the equations:

5X = 420
5X = 560

Solving for X: