Questions

If the first term (a = 79), the last term (b = 7), and the sum is 1118, then find the number of terms and common difference.

If the first term (a = 3), the last term (b = 90), and the sum is 1395, then find the number of terms and common difference.

If n(X) = 48, n(Y) = 51, n(Z) = 40, n(X $\rm \cap$ Y) = 11, n(Y $\rm \cap$ Z) = 10, n(Z $\rm \cap$ X) = 9, n(X $\rm \cap$ Y $\rm \cap$ Z) = 4 and $\rm n( \overline{X \cup Y \cup Z } )$ = 7, then

1. Find the value of $\rm n(X \cup Y \cup Z)$.
2. Find the value of n(U).
3. Show the information in a Venn diagram.
4. Find the cardinality of a set that is formed by the elements that are exactly in two of the sets X, Y, and Z. 

If n(A) = 14, n(B) = 13, n(C) = 22, n(A $\rm \cap$ B $\rm \cap$ C) = 4, n(A $\rm \cap$ B) = 4, n(B $\rm \cap$ C) = 9, n(C $\rm \cap$ A) = 11, and n ($\rm \overline { A \cup B \cup C }$) = 4, then

1. Find the value of n(U).
2. Find the value of n($\rm A \cup B \cup C$).
3. Show the information in a Venn diagram.
4. Is A a subset of (B $\rm \cup$ C)? Give reason.

If n(A) = 36, n(B) = 36, n(A $\rm \cap$ B) = 15, n(A $\rm \cap$ C) = 15, n(B $\rm \cap$ C) = 12, n(A $\rm \cup$ B $\rm \cup$ C) = 66, then

1. Find the value of $\rm n_{o}(C)$.
2. Find the value of $\rm n(C)$.
3. Find the value of $\rm n(A \cup C)$.
4. Present in a Venn diagram.

If n(A) = 12, n(B) = 12, n(A $\rm \cap$ B) = 5, n(A $\rm \cap$ C) = 3, n(B $\rm \cap$ C) = 4, n(A $\rm \cap$ B $\rm \cap$ C) = 2 and n(A $\rm \cup$ B $\rm \cup$ C) = 20, then

1. Find the value of $\rm n_{o}(C)$.
2. Find the value of $\rm n(C)$.
3. Find the value of $\rm n(A \cup C)$.
4. Present in a Venn diagram.

A, B and C are the subsets of the universal set U. If n(U) = 100, n(A) = 60, n(B) = 45, n(C) = 30, n(A $\rm \cap$ B) = 25, n(B $\rm \cap$ C) = 20, n(C $\rm \cap$ A) = 15, n(A $\rm \cup$ B $\rm \cup$ C) = 85.

1. The sets A, B, and C are overlapping sets. Give reason.
2. Find the value of n(A $\rm \cap$ B $\rm \cap$ C).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the given information in a Venn diagram.

Sets A, B, and C are the subsets of the universal set U. If n(U) = 300, n(A) = 100, n(B) = 90, n(C) = 110, n(A$\rm \cap$B) = 60, n(B$\rm \cap$C) = 40, n(C$\rm \cap$A) = 45, and n(A $\rm \cup$ B $\rm \cup$ C) = 200.

1. The sets A, B, and C are overlapping sets. Give reason.
2. Find the value of n(A $\rm \cap$ B $\rm \cap$ C).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the given information in a Venn diagram.

If n(A) = 48, n(B) = 51, n(C) = 40, n(A $\rm \cap$ B) = 11, n(B $\rm \cap$ C) = 10, n(C $\rm \cap$ A) = 9, n(A $\rm \cap$ B $\rm \cap$ C) = 4 and n(U) = 120,

1. Are the sets A, B, and C overlapping sets? Give reason.
2. Find the value of n($\rm A \cup B\cup C$).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the above information in a Venn diagram.

If the sum of the first 10 terms of an arithmetic series is 27.5 and the $\rm 10^{th}$ term of the series is 5, determine the value of its first term and common difference.