Question
Download Solution PDF\(\rm \sum\limits_{n=1}^{8n+7} i^n\) किसके बराबर है, जहाँ i = √-1 है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFप्रयुक्त सूत्र:
आयोटा की घात।
i2 = -1, i3 = -i, i4 = 1,
i5 = i, i6 = -1, i7 = -i......और आगे भी इसी तरह।
⇒ i1 + i2 + i3 + i4 = i -1 - i + 1 = 0
गणना:
⇒ i1 + i2 + i3 + i4 + i5 + i6 + i7 + ..... + i8n+7
⇒ i1 + i2 + i3 + i4 + i5 + i6 + i7 + ..... + i8n + i8n+1 + i8n+2 + .... + i8n+6 + i8n+7
⇒ (i -1 - i + 1) + (i -1 - i + 1) + .... + (i8n+1 + i8n+2 + i8n+3 + i8n+4 ) + (i8n+5 + i8n+6 + i8n+7)
⇒ i8n+5 + i8n+6 + i8n+7 = i1 + i2 + i3 = i -1 - i = -1
∴ \(\rm \sum\limits_{n=1}^{7} i^n\) का मान - 1 है।
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