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The sum S_n of the first n terms of an arithmetic series is given by $\"\"$ , where a₁ = first term, a_n = last term.

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1).

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Sum of the first half of the terms $\"\"$ .

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Sum of the second half of the terms $\"\"$ .

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Sum of the first term and last term $\"\"$ .

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Total sum = Sum of the first half of the terms + Sum of the first half of the terms.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The number of terms in the given arithmetic progression is 6.

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2).

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First term of the geometric progression (a ) = - 8.

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Last term of the geometric progressin $\"image\"$ .

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Substitute a = - 8 in $\"image\"$ .

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$\"image\"$ .

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Sum of geometric progression $\"image\"$ .

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Substitute a = - 8 and $\"image\"$ in $\"image\"$ .

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$\"image\"$

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$\"image\"$

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$\"image\"$

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$\"image\"$

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$\"image\"$ .

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To find n, substitute the value r = - 1 / 2 in $\"image\"$ .

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$\"image\"$

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$\"image\"$

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⇒ n = 4.

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The number of terms of the geometric progression is 4.

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