linto asked in Science & MathematicsMathematics · 6 months ago

# Determine the sum of the first 100 numbers. (arithmetic sequence)?

In an arithmetic sequence, the first three numbers are 100, 96 and 92.

Determine the sum of the first 100 numbers.

I only know that the answer is -9800, can someone show how I can get this answer?

Relevance
• Mike G
Lv 7
6 months ago

a = 100

d = -4

100th term = 100-4*99 = -396

Average term = (100-396)/2 = -98

Sum = -98*100 = -9800

• Como
Lv 7
6 months ago

Sn = (n/2) [ 2a + (n - 1) d ]

S100 = 50 [ 2 + 99 x 1 ]

S 100 = 50 x 101

S 100 = 5050

• sepia
Lv 7
6 months ago

100 + 96 + 92 + ...... + (100 - 4*99)

= 100/2(100 + 100 - 4*99)

= 50(200 - 396)

= 100(-196)/2

= -19600/2

= -9800

• 6 months ago

S(n) = n/2(2a + (n-1)d) is the general equation

where

n = 100

a = 100

d = -4

Substituting

S(100) = 100/2(2(100) + ( 100-1)(-4))

S(100) = 50(200 + 99(-4)

S(100) = 50(200 - 396)

S(100) = 50(-196)

S(100) = -9800 As required.

• 6 months ago

Sn = (n/2)[2a + (n - 1)d]....where a is the first term, d is the common difference and n is the number of terms

i.e. S₁₀₀ = (100/2)[200 - 4(99)]

=> S₁₀₀ = 50(-196)

so, -9800

:)>

• TomV
Lv 7
6 months ago

A(n) = 100 - 4(n-1)

A(1) = 100

A(100) = 100 - 396 = -296

Σ = number of terms multiplied by the average value of each term:

= 100(A(1) + A(100))/2

= 100(100-296)/2

= 50(-196)

Ans: Σ = -9800

Or:

Σ(104-4n) = 100(104) - 4Σn

= 10400 - 400(101)/2

= 10400 - 20200

= -9800

Either way the answer is the same, -9800

• 6 months ago

S[n] = (n/2) * (t[1] + t[n])

This holds true for all arithmetic sequences

S[100] = (100/2) * (t[1] + t[100])

S[100] = 50 * (t[1] + t[100])

The first term is 100, or 100 - 0, or 104 - 4, or 104 - 4 * 1

The 2nd term is 96, or 100 - 4, or 104 - 8, or 104 - 4 * 2

The 3rd term is 92, or 104 - 12, or 104 - 4 * 3

See a pattern?

t[1] = 100

t[100] = 104 - 4 * 100 = 104 - 400 = -296

50 * (100 + (-296)) =>

50 * (-196) =>

50 * 2 * (-98) =>

-98 * 100 =>

-9800