, 04 Jun 2021
Maths Solutions 4
Useful Formulas
Factoring Formulas
2
∎ �+� = �2 + 2�� + �2
2
∎ �−� = �2 − 2�� + �2
∎ �2 − �2 = � + � � − �
∎ �3 − �3 = � − � �2 + �� + �2
∎ �3 + �3 = � − � �2 − �� + �2
Quadratic Formulas
−� ± �2 − 4��
∎ If ��2 + �� + � = 0 then �=
2�
Logarithmic Functions
∎ log� � = � ⇒ � = �� � > 0, � ≠ 1 & � ≠ 0
�
∎ log� = log� � − log� � � > 0, � ≠ 1, � > 0 & � ≠ 0
�
∎ log� � ∙ � = log� � + log� � � > 0, � ≠ 1 & � ∙ � > 0
log� �
∎ log� � = � > 0, � ≠ 1, � > 0, � > 0 & � ≠ 1
log� �
∎ ln � = log� � � ≠ 0 & � = ����������� ��������
∎ log � = log10 � �≠0
∎ log� �� = � ∙ log� � � > 0, � ≠ 1 & � > 0
∎ �log� � = �log� � � > 0, � ≠ 1, � > 0 & � > 0
50 Math Problems With Solutions: Algebra 1
, 04 Jun 2021
Maths Solutions 5
Sequence & Series
�
∎ 1 = 1 + 1 + 1 + 1 + . . . . . . . . . . . � ����� = �
�=0
�
� �+1
∎ � = 1 + 2 + 3 + 4 + ...........� =
2
�=0
�
� � + 1 2� + 1
∎ �2 = 12 + 22 + 32 + 42 + . . . . . . . . . . . + �2 =
6
�=0
� 2
3 3 3 3 3
� �+1 3
∎ � = 1 + 2 + 3 + 4 +........... + � =
2
�=0
Arithmetic Progression ��
Let � − First term & � − common difference
∎ � th term of an ��, �� = � + � − 1 �
�
∎ Sum of � terms of an ��, �� = � + ��
2
�
�� = 2� + � − 1 �
2
Geometric Progression ��
Let � − First term & � − common ratio
∎ � th term of an ��, �� = ���−1
� 1 − ��
∎ Sum of � terms of an ��, �� =
1−�
�
∎ Sum of ∞ terms of an ��, �∞ = , � <1
1−�
50 Math Problems With Solutions: Algebra 1
, 04 Jun 2021
Maths Solutions 6
Exponential Series
1 2 3 4
∎ �=1+ + + + +.........∞
1! 2! 3! 4!
� �2 �3 �4
∎ �� = 1 + + + + +.........∞
1! 2! 3! 4!
�� + �−� �2 �4 �6
∎ = 1 + + + +.........∞
2 2! 4! 6!
�� − �−� � �3 �5 �7
∎ = + + + +.........∞
2 1! 3! 5! 7!
∞ ∞
1 1
∎ �= = � = 1, 2, 3. . . .
�! �−� !
�=0 �=0
∞
1 1 2 3 4
∎ = + + + + .........∞ = � − 1
�! 1! 2! 3! 4!
�=1
∞
1 2 3 4
∎ = + + + .........∞ = � − 2
�! 2! 3! 4!
�=2
∞
1 1 2 3 4
∎ = + + + + .........∞ = � − 1
� + 1 ! 1! 2! 3! 4!
�=0
∞
1 2 3 4
∎ = + + + .........∞ = � − 2
� + 2 ! 2! 3! 4!
�=0
∞
1 1 1 1 � + �−1
∎ = + + +.........∞ =
2�! 2! 4! 6! 2
�=0
∞
1 1 1 1 1 � − �−1
∎ = + + + +.........∞ =
2� − 1 ! 1! 3! 5! 7! 2
�=0
50 Math Problems With Solutions: Algebra 1
Maths Solutions 4
Useful Formulas
Factoring Formulas
2
∎ �+� = �2 + 2�� + �2
2
∎ �−� = �2 − 2�� + �2
∎ �2 − �2 = � + � � − �
∎ �3 − �3 = � − � �2 + �� + �2
∎ �3 + �3 = � − � �2 − �� + �2
Quadratic Formulas
−� ± �2 − 4��
∎ If ��2 + �� + � = 0 then �=
2�
Logarithmic Functions
∎ log� � = � ⇒ � = �� � > 0, � ≠ 1 & � ≠ 0
�
∎ log� = log� � − log� � � > 0, � ≠ 1, � > 0 & � ≠ 0
�
∎ log� � ∙ � = log� � + log� � � > 0, � ≠ 1 & � ∙ � > 0
log� �
∎ log� � = � > 0, � ≠ 1, � > 0, � > 0 & � ≠ 1
log� �
∎ ln � = log� � � ≠ 0 & � = ����������� ��������
∎ log � = log10 � �≠0
∎ log� �� = � ∙ log� � � > 0, � ≠ 1 & � > 0
∎ �log� � = �log� � � > 0, � ≠ 1, � > 0 & � > 0
50 Math Problems With Solutions: Algebra 1
, 04 Jun 2021
Maths Solutions 5
Sequence & Series
�
∎ 1 = 1 + 1 + 1 + 1 + . . . . . . . . . . . � ����� = �
�=0
�
� �+1
∎ � = 1 + 2 + 3 + 4 + ...........� =
2
�=0
�
� � + 1 2� + 1
∎ �2 = 12 + 22 + 32 + 42 + . . . . . . . . . . . + �2 =
6
�=0
� 2
3 3 3 3 3
� �+1 3
∎ � = 1 + 2 + 3 + 4 +........... + � =
2
�=0
Arithmetic Progression ��
Let � − First term & � − common difference
∎ � th term of an ��, �� = � + � − 1 �
�
∎ Sum of � terms of an ��, �� = � + ��
2
�
�� = 2� + � − 1 �
2
Geometric Progression ��
Let � − First term & � − common ratio
∎ � th term of an ��, �� = ���−1
� 1 − ��
∎ Sum of � terms of an ��, �� =
1−�
�
∎ Sum of ∞ terms of an ��, �∞ = , � <1
1−�
50 Math Problems With Solutions: Algebra 1
, 04 Jun 2021
Maths Solutions 6
Exponential Series
1 2 3 4
∎ �=1+ + + + +.........∞
1! 2! 3! 4!
� �2 �3 �4
∎ �� = 1 + + + + +.........∞
1! 2! 3! 4!
�� + �−� �2 �4 �6
∎ = 1 + + + +.........∞
2 2! 4! 6!
�� − �−� � �3 �5 �7
∎ = + + + +.........∞
2 1! 3! 5! 7!
∞ ∞
1 1
∎ �= = � = 1, 2, 3. . . .
�! �−� !
�=0 �=0
∞
1 1 2 3 4
∎ = + + + + .........∞ = � − 1
�! 1! 2! 3! 4!
�=1
∞
1 2 3 4
∎ = + + + .........∞ = � − 2
�! 2! 3! 4!
�=2
∞
1 1 2 3 4
∎ = + + + + .........∞ = � − 1
� + 1 ! 1! 2! 3! 4!
�=0
∞
1 2 3 4
∎ = + + + .........∞ = � − 2
� + 2 ! 2! 3! 4!
�=0
∞
1 1 1 1 � + �−1
∎ = + + +.........∞ =
2�! 2! 4! 6! 2
�=0
∞
1 1 1 1 1 � − �−1
∎ = + + + +.........∞ =
2� − 1 ! 1! 3! 5! 7! 2
�=0
50 Math Problems With Solutions: Algebra 1
