- WE'RE GOING TO WRITE THE GIVEN SUMS
USING SIGMA NOTATION OR SUMMATION NOTATION.
WHEN WE HAVE A SUM WRITTEN USING SIGMA NOTATION
OR THIS FORM HERE,
A SUB I IS THE FORMULA THAT GENERATES THE TERMS
IN THE SERIES WRITTEN USING THE VARIABLE, I,
WHERE THE VARIABLE, I, STARTS AT THE VALUE OF M,
INCREASES BY 1 UNTIL IT REACHES THE VALUE OF N.
SO TO WRITE THIS 1st SUM USING SIGMA OR SUMMATION NOTATION,
WE FIRST WANT TO LOOK FOR A PATTERN
SO THAT WE CAN WRITE A FORMULA TO GENERATE THESE TERMS.
TO DO THIS, IT'S OFTEN HELPFUL TO WRITE THE TERMS
IN FACTORED FORM OR AT LEAST AS A PRODUCT.
AND FOR THIS EXAMPLE, SINCE WE HAVE AN ARITHMETIC SERIES,
IT'S HELPFUL TO WRITE 2 AS 1 x 2, 4 AS 2 x 2,
6 AS 3 x 2, 8 AS 4 x 2, 10 AS 5 x 2,
AND IF WE CONTINUE THIS PATTERN,
FOR 120 WE'D WANT A FACTOR OF 2.
SO THIS WOULD BE 60 x 2.
NOW LET'S LOOK FOR A PATTERN.
NOTICE EACH OF THESE PRODUCTS CONTAINS A FACTOR OF 2,
WHICH WOULD BE A CONSTANT IN OUR FORMULA, A SUB I.
AND THEN LOOKING AT THE OTHER FACTORS,
NOTICE HOW WE HAVE 1, 2, 3, 4, 5, ALL THE WAY OUT TO 60.
AND NOTICE HOW THIS FACTOR IS INCREASING BY 1
FOR EACH PRODUCT,
WHICH IS GOOD NEWS, BECAUSE WE CAN USE THE INDEX, I
FOR THE FIRST FACTORS,
AND THE SECOND FACTOR IS A CONSTANT,
WHICH MEANS WE'D HAVE THE SUMMATION OF I x 2 OR 2I.
AND AGAIN, THIS 2 HERE IS THE SECOND FACTOR,
AND THIS I IS OUR INDEX THAT INCREASES BY 1 EACH TIME.
SO I IS GOING TO START AT 1 AND STOP AT 60.
THIS WOULD BE THE SUM OR THE SERIES WRITTEN
USING SUMMATION NOTATION.
OF COURSE, WE CAN CHECK THIS.
NOTICE WHEN I = 1, WE'D HAVE 2 x 1 OR 2.
WHEN I = 2, WE'D HAVE 2 x 2 OR 4.
WHEN I = 3, WE'D HAVE 2 x 3 OR 6, AND SO ON
ALL THE WAY OUT TO I = 60 WHERE WE'D HAVE 2 x 60 OR 120.
NOW, LOOKING AT THE SECOND EXAMPLE,
AGAIN, OUR GOAL HERE IS TO FIND A PATTERN
SO WE CAN WRITE A FORMULA TO GENERATE THESE FRACTIONS,
AND THE INFORMATION PROVIDED IS VERY HELPFUL.
NOTICE HOW THE NUMERATOR STARTS AT 1,
INCREASES BY ONE EACH TIME, ALL THE WAY OUT TO 15.
SO WE CAN USE THIS AS OUR INDEX OR THE VARIABLE
IN THE FORMULA,
AND THEN LOOKING AT THE DENOMINATOR,
NOTICE HOW THE FIRST FACTOR IS ALWAYS ONE LARGER
THAN THE NUMERATOR, WHICH WOULD BE OUR INDEX, I.
AND THEN THE SECOND FACTOR IS ALWAYS TWO LARGER
THAN THE NUMERATOR OR OUR INDEX,
WHICH MEANS WE CAN WRITE THIS SUM USING SIGMA NOTATION
OR SUMMATION NOTATION
USING THE FORMULA I FOR THE NUMERATOR.
AND THEN FOR THE DENOMINATOR,
SINCE THE GREEN FACTOR OR THE FIRST FACTOR IS ALWAYS
ONE LARGER THAN THE NUMERATOR,
WE CAN WRITE THE FIRST FACTOR AS I + 1,
AND THEN SINCE THE SECOND FACTOR IS ALWAYS TWO LARGER
THAN THE NUMERATOR OR THE INDEX,
THEN THE SECOND FACTOR WOULD BE I + 2.
LET'S GO AHEAD AND CENTER THIS I HERE.
NOW, WE CAN'T FORGET ABOUT THE INDEX OR THE VALUES OF I
THAT WILL GENERATE THESE TERMS.
THE LOWER BOUND FOR "I,"
WHICH IS WHERE "I" STARTS, WILL BE AT 1.
AND THE UPPER BOUND WHERE "I" WILL STOP IS AT I = 15.
THIS WOULD BE THE SIGMA OR SUMMATION NOTATION
FOR THE GIVEN SUM.
NOTICE WHEN I = 1, WE'D HAVE 1/2 x 3,
WHICH IS OUR FIRST FRACTION.
WHEN I = 2, WE'D HAVE 2/3 x 4,
WHICH WOULD BE OUR SECOND FRACTION, AND SO ON.
SO THE CHALLENGING PART ABOUT WRITING A SUM OR A SERIES
USING SIGMA NOTATION IS ANALYZING THE TERMS
TO SEE THE PATTERN THAT WE CAN USE
TO WRITE THE FORMULA TO GENERATE EACH TERM.
I HOPE YOU FOUND THIS HELPFUL.