Question

The sum of the first nine terms is 117. the sum of the next four terms is 91. find the first term and the common difference

Answer 1

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The sum of the first nine terms is 117. the sum of the next four terms is 91. find the first term and the common difference

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$\sf\small\underline\red{Given \: data:-}$

Sum of first 6 terms of arithmetic progression

Sum of first 12 terms of arithmetic progression

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$\sf\small\underline\red{To \: find:-}$

the sum of first 30 terms of arithmetic progression

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$\sf\small\underline\red{Solution:-}$

The formula to find the sum of first n terms of arithmetic progression is

Where n is the number of terms

a is the first term of the progression d is the common difference For the sum of first 6 terms,

39 = 2a +5d _________________(1)

For the sum of first 12 terms,

81 = (2a + 11 d) _______________(2)

Subtract (2) from (1)

we get, 6 d = 42

d = 7

Substitute the value of 'd' in equation (2)

$\bf 2 a + 11 (7) = 81 \\ \\ \bf2 a +77 = 81 \\ \\ \bf2 a = 81-77 \\ \\ \bf2a = 4 \\ \\ \bf \red{a = 2}$

Therefore the sum of first 30 terms of the arithmetic progression is 3105 when the sum of first 6 terms is 117 and sum of first 12 terms is 486.

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Answer 2

Answer:

Form simultaneous equations from Sum of n terms formula

Step-by-step explanation:

(9/2)(2(a)+d(9-1))=117

simplify the above equation

Sum of next 4 terms=(Sum of first 13 terms)-(Sum of first 9 terms)

91=(11/2)(2a+d(11-1))-(9/2)(2a+d(9-1))

simplify the above equation

You will now have 2 simultaneous equations, you can now solve them using substitution or elimination.