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Calculator Technique for Arithmetic Progression
Problem: Arithmetic Progression
The 6 th term of an arithmetic progression is 12 and the 30th term is 180.
1. What is the common difference of the sequence?
2. Determine the first term?
3. Find the 52 nd term.
4. If the nth term is 250, find n.
5. Calculate the sum of the first 60 terms.
6. Compute for the sum between 12th and 37th terms, inclusive.
Put your calculator to Linear Regression in STAT mode:
MODE → 3:STAT → 2:A+BX and input the coordinates.
Among the many
STAT types, why A+BX ?
The formula a n = a m + (n – m)d is linear in n. In calculator, we input n at X column and a n at Y column. Thus our X is linear representing the variable n in the formula.
X (for n)= 6, 30
Y (for a n)= 12, 180
To find the first term:
AC → 1 SHIFT → 1[STAT] → 7:Reg → 5:y-caret and calculate
1y-caret , be sure to place 1 in front of y-caret.
1y-caret = -23 → answer for the first term.
To find the 52nd term, and again
AC → 52 SHIFT → 1[STAT] → 7:Reg → 5:y-caret and make sure you place 52 in front of y-caret.
52y-caret = 334 → answer for the 52nd term.
To find n for a n = 250,
AC → 250 SHIFT → 1[STAT] → 7:Reg → 4:x-caret
250x-caret = 40 → answer for n
To find the common difference, solve for any term adjacent to a given term, say 7th term because the 6th term is given then do 7y-caret – 12 = 7 for d. For some fun, randomly subtract any two adjacent terms like 18y-caret –
17y-caret , etc. Try it!
answer.Sum of Arithmetic Progression by Calculator
Sum of the first 60 terms:
AC → SHIFT → log[Σ] → ALPHA → )[X] → SHIFT → 1[STAT] → 7:Reg → 5:y-caret → SHIFT → )[,] → 1 → SHIFT → )[,] → 60 → )
The calculator will display Σ(Xy-caret, 1, 60) then press [=].
Σ(Xy-caret, 1, 60) = 11010 ←
answer
Sum from 12th to 37th terms,
Σ(Xy-caret, 12, 37) = 3679 ←
answer.
Another way to solve for the sum is to use the Σ calculation outside the STAT mode. The concept is to add each term in the progression. Any term in the progression is given by a n = a 1 + (n – 1)d. In this problem, a 1 = -23 and d = 7, thus, our equation for a n is a n = -23 + (n – 1)(7).
Reset your calculator into general calculation mode: MODE → 1:COMP then SHIFT → log .
Sum of first 60 terms:
Press shift log at the right top of Casio fx991es to bring out summation symbol, then input the values, at the top input 60 and your x=1 at the right hand side input these values (-23 + (ALPHA X – 1) × 7) = 11010
Or you can do
Press shift log at the right top of Casio fx991es to bring out summation symbol, at the top input 59 and your x=0 and at the right hand side input these values (-23 + 7 ALPHA X) = 11010
which yield the same result.
Sum from 12th to 37th terms
Press shift log at the right top of Casio fx991es to bring out summation symbol, at the top input 37 and your x=12 and at the right hand side input these values (-23 + (ALPHA X – 1) × 7) = 3679
Or you may do
Press shift log at the right top of Casio fx991es to bring out summation symbol, at the top input 36 and your x=11 and at the right hand side input (-23 + 7 ALPHA X) = 3679
which yield the same result.
Calculator Technique for Geometric Progression
Problem…………….
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