Some addition patterns are given below. You may practise these. If you find any irregularity, you may send suggestions. I think, you will enjoy these.
v Adding even numbers from two through
a selected 2-digit even number
Divide the even number by 2 (or multiply by 1/2).
Multiply this result by the next number.
Example:
If the 2-digit even number selected is 24:
Divide 24 by 2 (24/2 = 12) or multiply by 1/2 (1/2 × 24 = 12).
The next number is 13; 12 × 13 = 156.
Ways to multiply 13 by 12:
· Square 12, then add 12: 12 × 12 = 144, 144 + 12 = 156.
· Multiply left to right. 12 × 13 can be done in steps: 12 × (10+3) = (12 × 10) + (12 × 3) = 120 + 36 = 156.
So the sum of all the even numbers from two through 24 is 156.
See the pattern?
If the 2-digit even number selected is 42:
Divide 42 by 2 (42/2 = 21) or multiply by 1/2 (1/2 × 42 = 21).
The next number is 22; 21 × 22 = 462.
Ways to multiply 22 by 21:
· Square 21, then add 21: 21 × 21 = 441, 441 + 21 = 462.
· Multiply left to right. 21 × 22 can be done in steps: 21 × 22 = (21 × 20) + (21 x 2) = 420 + 42 = 462.
So the sum of all the even numbers from two through 42 is 462.
v Adding the digits of the square of repeating ones
Square a repeating ones number (111, 11111, etc.).
Add the digits of the square.
Example:
If the number selected is 1111:
Square the number: 1111 × 1111 = 1234321
Add the digits of the square:
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
The answer is the square of the number of ones in 1111.
See the pattern?
If the number selected is 111111:
Square the number: 111111 × 111111 = 12345654321
Add the digits of the square:
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36
The answer is the square of the number of ones in 111111.
v Adding a sequence of consecutive odd numbers
Choose a 2-digit odd number. Add all the odd numbers starting with one through this 2-digit number:
Add one to the 2-digit number.
Divide this sum by 2 (take half of it).
Square this number. This is the sum of all odd numbers from 1 through the 2-digit number chosen.
Example:
If the 2-digit odd number selected is 35:
35+1 = 36 (add 1).
36/2 = 18 (divide by 2) or 1/2 × 36 = 18 (multiply by 1/2).
18 × 18 = 324 (square 18): 18 × 18 = (20 - 2)(18) = (20 × 18) - (2 × 18) = 360 - 36 = 360 - 30 - 6 = 324.
So the sum of all the odd numbers from one through 35 is 324.
See the pattern?
If the 2-digit odd number selected is 79:
79+1 = 80 (add 1).
80/2 = 40 (divide by 2) or 1/2 × 80 = 40 (multiply by 1/2).
40 × 40 = 1600 (square 40).
So the sum of all the odd numbers from one through 79 is 1600.
v Adding consecutive numbers between two numbers
Choose two 2-digit numbers less than 20 (no limits for experts).
Add all the numbers between them:
Add the numbers;
Subtract the numbers and add 1;
Multiply half the sum by this difference + 1, OR
Multiply the sum by half the difference + 1.
Example:
If the two numbers selected are 6 and 19:
Add the numbers: 6 + 19 = 25.
Subtract the numbers: 19 - 6 = 13. Add 1: 13 + 1 = 14.
Multiply 25 by half of 14: 25 × 7 = 175.
So the sum of the numbers from 6 through 19 is 175.
See the pattern?
If the two numbers selected are 4 and 18:
Add the numbers: 4 + 18 = 22.
Subtract the numbers: 18 - 4 = 14. Add 1: 14 + 1 = 15.
Multiply half of 22 by 15: 11 × 15 = 165 (10 × 15 + 15).
So the sum of the numbers from 4 through 18 is 165.
Practice with examples restricted to less than 20, then advance to larger numbers. You will impress your friends when you beat the calculator.
v Add a sequence from one to a selected 2-digit number
Choose a 2-digit number.
Multiply the 2-digit number by half the next number, or
Multiply half the 2-digit number by the next number.
Example:
If the 2-digit even number selected is 51:
The next number is 52. Multiply 51 times half of 52.
51 × 26: (50 × 20) + (50 × 6) + 1 × 26) =
1000 + 300 + 26 = 1326
So the sum of all numbers from 1 through 51 is 1326.
See the pattern?
If the 2-digit even number selected is 34:
The next number is 35. Multiply half of 34 × 35.
17 × 35: (10 × 35) + (7 × 30) + (7 × 5) =
350 + 210 + 35 = 560 + 35 = 595
So the sum of all numbers from 1 through 34 is 595.
With some multiplication practice you will be able to find these sums of sequential numbers easily and faster than someone using a calculator!
v Add a sequence from one to a selected 1-digit number and back
Choose a 1-digit number.
Square it.
Example:
If the 1-digit number selected is 7:
To add 1 + 2 + 3 + 4 + 5 + 6 + 7 + 6 + 5 + 4 + 3 + 2 + 1
Square 7: 49
So the sum of all numbers from 1 through 7 and back is 49.
See the pattern?
If the 1-digit number selected is 9:
To add 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
Square 9: 81
So the sum of all numbers from 1 through 9 and back is 81.
You will certainly be faster with your answer than someone adding all the numbers up on a calculator!
v Add sequences of numbers in the 10's
Choose a 2-digit number in the 10's. To add all the 10's from 10 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
The first digits = the number of terms
(+ carry). nbsp; X X _
Example:
If the 2-digit number in the 10's selected is 16:
(10 + 11 + 12 ... 16 + 15 + 14 ... 10)
Square the 2nd digit of the number:
6 × 6 = 36 (keep carry 3) _ _ 6
No. of terms = 2 × 2nd digit + 1:
2 × 6 + 1 = 13
No. of terms (+ carry): 13 + 3 = 16 1 6 _
So the sum of the sequence is 166.
See the pattern?
If the 2-digit number in the 10's selected is 18:
(10 + 11 + 12 + ... 18 + 17 + 16 + 15 ... 10)
Square the 2nd digit of the number:
8 × 8 = 64 (keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1:
2 × 8 + 1 = 17
No. of terms (+ carry): 17 + 6 = 23 2 3 _
So the sum of the sequence is 234.
v Add sequences of numbers in the 20's
Choose a 2-digit number in the 20's. To add all the 20's
from 20 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
Multiply the number of terms by 2 (+ carry). X X _
Example:
If the 2-digit number in the 20's selected is 23:
(20 + 21 + 22 + 23 + 22 + 21 + 20)
Square the 2nd digit of the number: 3 × 3 = 9 _ _ 9
No. of terms = 2 × 2nd digit + 1: 2 × 3 + 1 = 7
2 × no. of terms: 2 × 7 = 14 1 4 _
So the sum of the sequence is 149.
See the pattern?
If the 2-digit number in the 20's selected is 28:
(20 + 21 + 22 ... + 28 + 27 + ... 22 + 21 + 20)
Square the 2nd digit of the number: 8 × 8 = 64
(keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1: 2 × 8 + 1 = 17
2 × no. of terms (+ carry): 2 × 17 + 6 = 40 4 0 _
So the sum of the sequence is 404.
v Add sequences of numbers in the 30's
Choose a 2-digit number in the 30's. To add all the 30's
from 30 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
Multiply the number of terms by 3 (+ carry) X X _
Example:
If the 2-digit number in the 30's selected is 34:
(30 + 31 + 32 + 33 + 34 + 33 + 32 + 31 + 30)
Square the 2nd digit of the number:
4 × 4 = 16 (keep carry 1) _ _ 6
No. of terms = 2 × 2nd digit + 1:
2 × 4 + 1 = 9
3 × no. of terms: 3 × 9 + 1 = 28 2 8 _
So the sum of the sequence is 286.
See the pattern?
If the 2-digit number in the 30's selected is 38:
(30 + 31 + 32 + ... + 38 + 37 + ... 32 + 31 + 30)
Square the 2nd digit of the number:
8 × 8 = 64 (keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1:
2 × 8 + 1 = 17
3 × no. of terms: 3 × 17 + 6 = 51 + 6 = 57 5 7 _
So the sum of the sequence is 574.
v Add sequences of numbers in the 40's
Choose a 2-digit number in the 40's. To add all the 40's
from 40 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
Multiply the number of terms by 4 (+ carry) X X _
Example:
If the 2-digit number in the 40's selected is 43:
(40 + 41 + 42 + 43 + 42 + 41 + 40)
Square the 2nd digit of the number:
3 × 3 = 9 _ _ 9
No. of terms = 2 × 2nd digit + 1:
2 × 3 + 1 = 7
4 × no. of terms: 4 × 7 = 28 2 8 _
So the sum of the sequence is 289.
See the pattern?
If the 2-digit number in the 40's selected is 48:
(40 + 41 + 42 + ... + 48 + 47 + ... 42 + 41 + 40)
Square the 2nd digit of the number:
8 × 8 = 64 (keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1:
2 × 8 + 1 = 17
4 × no. of terms: 4 × 17 + 6 = 40 + 28 + 6 =
68 + 6 = 74 7 4 _
So the sum of the sequence is 744.
v Add sequences of numbers in the 50's
Choose a 2-digit number in the 50's. To add all the 50's
from 50 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
Multiply the number of terms by 5. X X _
Example:
If the 2-digit number in the 50's selected is 53:
(50 + 51 + 52 + 53 + 52 + 51 + 50)
Square the 2nd digit of the number: 3 × 3 = 9 _ _ 9
No. of terms = 2 × 2nd digit + 1: 2 × 3 + 1 = 7
5 × no. of terms: 5 × 7 = 35 3 5 _
So the sum of the sequence is 359.
See the pattern?
If the 2-digit number in the 50's selected is 57:
(50 + 51 + 52 ... + 57 + 56 + ... 52 + 51 + 50)
Square the 2nd digit of the number: 7 × 7 = 49
(keep carry 4) _ _ 9
No. of terms = 2 × 2nd digit + 1: 2 × 7 + 1 = 15
5 × no. of terms (+ carry): 5 × 15 + 4 = 75 + 4
= 79 7 9 _
So the sum of the sequence is 799.
v Add sequences of numbers in the 60's
Choose a 2-digit number in the 60's. To add all the 60's
from 60 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
Multiply the number of terms by 6 (+ carry) X X _
Example:
If the 2-digit number in the 60's selected is 63:
(60 + 61 + 62 + 63 + 62 + 61 + 60)
Square the 2nd digit of the number:
3 × 3 = 9 _ _ 9
No. of terms = 2 × 2nd digit + 1:
2 × 3 + 1 = 7
6 × no. of terms: 6 × 7 = 42 4 2 _
So the sum of the sequence is 429.
See the pattern?
If the 2-digit number in the 60's selected is 68:
(60 + 61 + 62 + ... + 68 + 67 + ... 62 + 61 + 60)
Square the 2nd digit of the number:
8 × 8 = 64 (keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1:
2 × 8 + 1 = 17
6 × no. of terms: 6 × 17 + 6 = 42 + 1 =
102 + 6 = 108 1 0 8 _
So the sum of the sequence is 1084.
v Add sequences of numbers in the 70's
Choose a 2-digit number in the 70's. To add all the 70's
from 70 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1.
Multiply the number of terms by 7 (+ carry) X X _
Example:
If the 2-digit number in the 70's selected is 73:
(70 + 71 + 72 + 73 + 72 + 71 + 70)
Square the 2nd digit of the number:
3 × 3 = 9 _ _ 9
No. of terms = 2 × 2nd digit + 1:
2 × 3 + 1 = 7
7 × no. of terms: 7 × 7 = 49 4 9 _
So the sum of the sequence is 499.
See the pattern?
If the 2-digit number in the 70's selected is 78:
(70 + 71 + 72 + ... + 78 + 77 + ... 72 + 71 + 70)
Square the 2nd digit of the number:
8 × 8 = 64 (keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1:
2 × 8 + 1 = 17
7 × no. of terms: 7 × 17 + 6 = 49 + 6 =
119 + 6 = 125 1 2 5 _
So the sum of the sequence is 1254.
v Add sequences of numbers in the 80's
Choose a 2-digit number in the 80's. To add all the80's
from 80 up through this number and down from it:
Square the 2nd digit of the number
(keep the carry) _ _ X
The number of terms is 2 × the 2nd digit + 1
Multiply the number of terms by 8 (add the carry) X X _
Example:
If the 2-digit number in the 80's selected is 83:
(80 + 81 + 82 + 83 + 82 + 81 + 80)
Square the 2nd digit of the number: 3 × 3 = 9 _ _ 9
No. of terms = 2 × 2nd digit + 1: 2 × 3 + 1 = 7
8 × no. of terms: 8 × 7 = 56 5 6 _
So the sum of the sequence is 569.
See the pattern?
If the 2-digit number in the 80's selected is 88:
(80 + 81 + 82 ... + 88 + 87 + ... 83 + 82 + 81 + 80)
Square the 2nd digit of the number: 8 × 8 = 64
(keep carry 6) _ _ 4
No. of terms = 2 × 2nd digit + 1: 2 × 8 + 1
= 17
8 × no. of terms (+ carry): 8 × 17 + 6
= 80 + 56 + 6 = 136 + 6 = 42 4 2 _
So the sum of the sequence is 1424.
v Adding a series of doubles
Have a friend choose a a single digit number. (No restrictions for experts.)
Ask your friend to jot down a series of doubles (where the next term is always double the preceding one), and tell you the last term.
Ask your friend to add up all these terms.
You will give the answer before he or she can finish: The sum of all the terms of this series will be two times the last term minus the first term.
Example:
If the number selected is 9:
The series jotted down is: 9, 18, 36, 72, 144.
Two times the last term (144) minus the first (9):
2 × 144 = 288; 288 - 9 = 279.
So the sum of the doubles from 9 through 144 is 279.
See the pattern? Here's one for the experts:
The number selected is 32:
The series jotted down is: 64, 128, 256, 512.
Two times the last term (512) minus the first (64):
2 × 512 = 1024; 1024 - 32 = 1024 - 30 - 2 = 994 - 2 = 992.
So the sum of the doubles from 32 through 512 is 992.
Remember to subtract in steps from left to right.
With practice you will be expert in summing series.
v Adding a series of quadruples
Have a friend choose a single digit number. (No restrictions for experts.)
Ask your friend to jot down a series of quadruples (where the next term is always four times the preceding one), and tell you only the last term.
Ask your friend to add up all these terms.
You will give the answer before he or she can finish: The sum of all the terms of this series will be four times the last term minus the first term, divided by 3.
Example:
If the number selected is 5:
The series jotted down is: 5, 20, 80, 320, 1280.
Four times the last term (1280) minus the first (5):
4000 + 800 + 320 - 5 = 5120 - 5 = 5115
Divide by 3: 5115/3 = 1705
So the sum of the quadruples from 5 through 1280 is 1705.
See the pattern? Here's one for the experts:
The number selected is 32:
The series jotted down is: 32, 128, 512, 2048.
Four times the last term (2048) minus the first (32):
8000 + 160 + 32 - 32 = 8,160
Divide by 3: 8160/3 = 2720.
So the sum of the quadruples from 32 through 2048 is 2720.
Practice multiplying from left to right and dividing by 3. With practice you will be an expert quad adder.
v Adding a series of ten numbers
Have a friend choose and write down a single-digit number. (Two digits for experts.)
Ask your friend to name and note a third number by adding the first two.
Name a fourth by adding the second and third. Continue in this way, announcing each number, through ten numbers.
Ask your friend to add up the ten numbers. You will give the answer before he or she can finish:
The sum of all the terms of this series will be the seventh number multiplied by 11.
Example:
If the numbers selected are 7 and 4:
The series jotted down is: 4, 7, 11, 18, 29, 47, 76, 123, 199, 322.
The seventh number is 76. 11 × 76 = 836
(use the shortcut for 11: 7 is the first digit, 6 is the third digit;
the middle digit will be 7 + 6, and carry the 1: 836).
So the sum of the ten numbers is 836.
