a) Use this information to form an equation in a and d.

STEP 1: Identify the first term in the sequence, call this number a.

STEP 2: Calculate the common difference(d) of the sequence.

[Common difference = Succeeding(a+2d) - Preceding(a+d) term ]

STEP 3: Find the term position ( n ):

[EXAMPLE: 2,4,6,8,10….in AP

First term 2, term position (n) = 1

Second term 4, term position (n) = 2........

......... 30th term, term position (n) = 30.......]

STEP 4: Substitute all the values in the arithmetic formula

[FORMULA: Tn = a + (n - 1) d,

where Tn = nth term and a = first term. Here d = common difference, n = term position.]

STEP 5: Equate the nth term with the earnings of the month

Answer: a + 29d = £40.75...............................(1)

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A picker who works for all 30 days will earn a total of £1005

(b) Show that 15(a + 40.75) = 1005 ............................................(2)

STEP 1: Make sure you have an arithmetic sequence.

STEP 2: Identify the number of terms(n) in your sequence.

STEP 3: Identify the first(a_1) and last(a_n) terms in the sequence. Plug the values of n, a_1, and a_n into the formula.

[FORMULA: The formula for finding the sum of an arithmetic sequence(S_n) =(n/2)(a_1 + a_n).]

STEP 4: Simplify further. And verify

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(c) Hence find the value of a and the value of d.

STEP 1: Simplify the sum of an arithmetic sequence 15(a + 40.75) = 1005 [from equation (2)]

a + 40.75 = 1005/15

a + 40.75 = 67

a = 67 - 40.75

a = 26.25 pounds

STEP 2: Substitute "a" value in the nth term

a + 29d = 40.75 [from equation (1)]

==> 26.25 + 29d = 40.75

==> 29d = 40.75 - 26.25

29d = 14.5

d = (14.5)/29

d = 0.5 pounds