We know that that is lanthanides belongs to generally 57 to 71 .atomic no 63 also lies in lanthanides series.in D block group 3.y=actinides period 7and z=oxidation state =3
so x+y+z=3+7+3=13

Europium (atomic number 63)
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Europium belongs to the f-block and is a member of the lanthanide series. The atomic number of europium is 63, which means it has 63 protons in its nucleus. Protons determine the element's atomic number and its position on the periodic table.

Group number of Europium = x = ?

Period number of Americium (atomic number 95)
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Americium is a member of the actinide series and also belongs to the f-block. The atomic number of americium is 95, indicating that it has 95 protons.

Period number of Americium = y = ?

Most stable oxidation state shown by Gadolinium (atomic number 64)
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Gadolinium is also a member of the f-block and belongs to the lanthanide series. The atomic number of gadolinium is 64, which means it has 64 protons.

Most stable oxidation state shown by Gadolinium = z = ?

To find the values of x, y, and z, we need to analyze the periodic table.

Analyzing the Periodic Table
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The periodic table is divided into periods (rows) and groups (columns). The period number indicates the energy level or shell in which the element's valence electrons are located. The group number represents the number of valence electrons an element has.

Europium is in the lanthanide series, which is located in period 6. Therefore, the period number of europium is 6.

Americium is in the actinide series, which is located in period 7. Therefore, the period number of americium is 7.

Gadolinium is also in the lanthanide series and is located in period 6. Therefore, the period number of gadolinium is 6.

Determining the Group Number of Europium (x)
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The group number of europium can be determined by looking at its position in the periodic table. Europium is located in the f-block, specifically in group 3. Therefore, the group number of europium is 3.

Determining the Most Stable Oxidation State of Gadolinium (z)
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The most stable oxidation state shown by an element can vary. To determine the most stable oxidation state of gadolinium, we can refer to its position in the periodic table and its electron configuration.

Gadolinium has an atomic number of 64, which means it has 64 electrons. The electron configuration of gadolinium is [Xe] 4f^7 5d^1 6s^2. From this configuration, we can deduce that the most stable oxidation state of gadolinium is +3. In this oxidation state, gadolinium loses three electrons from its 4f orbital, leaving it with a stable configuration of [Xe] 4f^7.

Therefore, the most stable oxidation state shown by gadolinium is z = +3.

Final Values of x, y, and z
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From the analysis above, we can conclude that:

x (Group number of Europium) = 3
y (Period number of Americium) = 7
z (Most stable oxidation state shown by Gadolinium) = +3

Therefore, the correct answer is 13.